Left Termination of the query pattern qs_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

qs([], []).
qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs).
part(X1, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

qs(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: qs(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: qs(T1, T2), SCOPE: 1, CLAUSE: 0 | qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nqs(T1, T2) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: qs([], 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: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9))))\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 2 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(less(T22, T23), ','(part(T22, T24, X52, X53), ','(qs(.(T23, X52), X14), ','(qs(X53, X15), app(X14, .(T22, X15), T9)))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX14 is free; \nX15 is free; \nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: less(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(part(T22, T24, X52, X53), ','(qs(.(T23, X52), X14), ','(qs(X53, X15), app(X14, .(T22, X15), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX14 is free; \nX15 is free; \nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(T22, T23), SCOPE: 3, CLAUSE: 7 | less(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T22, T23), SCOPE: 3, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", 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: less(T38, T39)\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: part(T22, T24, X52, X53)\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(qs(.(T23, T45), X14), ','(qs(T46, X15), app(X14, .(T22, X15), T9)))\n\nT22 is ground\nT23 is ground\nT45 is ground\nT46 is ground\nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 2 | part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(less(T62, T63), part(T62, T64, X126, X127))\n\nT62 is ground\nT63 is ground\nT64 is ground\nX126 is free; \nX127 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: less(T62, T63)\n\nT62 is ground\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: part(T62, T64, X126, X127)\n\nT62 is ground\nT64 is ground\nX126 is free; \nX127 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 3\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: part(T84, T86, X177, X178)\n\nT84 is ground\nT86 is ground\nX177 is free; \nX178 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: qs(.(T23, T45), X14)\n\nT23 is ground\nT45 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(qs(T46, X15), app(T95, .(T22, X15), T9))\n\nT22 is ground\nT46 is ground\nT95 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: qs(T46, X15)\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(T95, .(T22, T98), T9)\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: qs(T46, X15), SCOPE: 5, CLAUSE: 0 | qs(T46, X15), SCOPE: 5, CLAUSE: 1\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: qs(T46, X15), SCOPE: 5, CLAUSE: 0\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: qs(T46, X15), SCOPE: 5, CLAUSE: 1\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(part(T103, T104, X226, X227), ','(qs(X226, X228), ','(qs(X227, X229), app(X228, .(T103, X229), X230))))\n\nT103 is ground\nT104 is ground\nX230 is free; \nX226 is free; \nX227 is free; \nX228 is free; \nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: part(T103, T104, X226, X227)\n\nT103 is ground\nT104 is ground\nX226 is free; \nX227 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(qs(T108, X228), ','(qs(T109, X229), app(X228, .(T103, X229), X230)))\n\nT103 is ground\nT108 is ground\nT109 is ground\nX230 is free; \nX228 is free; \nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: qs(T108, X228)\n\nT108 is ground\nX228 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(qs(T109, X229), app(T113, .(T103, X229), X230))\n\nT103 is ground\nT109 is ground\nT113 is ground\nX230 is free; \nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: qs(T109, X229)\n\nT109 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: app(T113, .(T103, T114), X230)\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 5 | app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 6\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 5\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 6\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: app(T138, .(T139, T140), X270)\n\nT138 is ground\nT139 is ground\nT140 is ground\nX270 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 5 | app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 6\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 5\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 6\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T170, .(T171, T172), T174)\n\nT170 is ground\nT171 is ground\nT172 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(part(T191, T193, X336, X337), ','(qs(X336, X14), ','(qs(.(T192, X337), X15), app(X14, .(T191, X15), T9))))\n\nT191 is ground\nT192 is ground\nT193 is ground\nX14 is free; \nX15 is free; \nX336 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: part(T191, T193, X336, X337)\n\nT191 is ground\nT193 is ground\nX336 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(qs(T197, X14), ','(qs(.(T192, T198), X15), app(X14, .(T191, X15), T9)))\n\nT191 is ground\nT192 is ground\nT197 is ground\nT198 is ground\nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: qs(T197, X14)\n\nT197 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(qs(.(T192, T198), X15), app(T202, .(T191, X15), T9))\n\nT191 is ground\nT192 is ground\nT198 is ground\nT202 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(qs([], X14), ','(qs([], X15), app(X14, .(T207, X15), T9)))\n\nT207 is ground\nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: qs([], X14)\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(qs([], X15), app(T210, .(T207, X15), T9))\n\nT207 is ground\nT210 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: qs([], X14), SCOPE: 8, CLAUSE: 0 | qs([], X14), SCOPE: 8, CLAUSE: 1\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: qs([], X14), SCOPE: 8, CLAUSE: 0\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: qs([], X14), SCOPE: 8, CLAUSE: 1\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: qs([], X15)\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: app(T210, .(T207, T213), T9)\n\nT207 is ground\nT210 is ground\nT213 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 94 [arrowhead = none ];
94 -> 2;

95 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 95 [arrowhead = none ];
95 -> 3;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 96 [arrowhead = none ];
96 -> 4;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 97 [arrowhead = none ];
97 -> 5;

98 [label="EVAL with clause\nqs(.(X9, X10), X11) :- ','(part(X9, X10, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(X9, X15), X11)))).\nand substitution\nX9 -> T6\nX10 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX11 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 98 [arrowhead = none ];
98 -> 7;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 99 [arrowhead = none ];
99 -> 8;

100 [label="BACKTRACK\nfor clause: qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 100 [arrowhead = none ];
100 -> 6;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 101 [arrowhead = none ];
101 -> 9;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 102 [arrowhead = none ];
102 -> 10;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 103 [arrowhead = none ];
103 -> 11;

104 [label="EVAL with clause\npart(X47, .(X48, X49), .(X48, X50), X51) :- ','(less(X47, X48), part(X47, X49, X50, X51)).\nand substitution\nT6 -> T22\nX47 -> T22\nX48 -> T23\nX49 -> T24\nT7 -> .(T23, T24)\nX50 -> X52\nX12 -> .(T23, X52)\nX13 -> X53\nX51 -> X53", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 104 [arrowhead = none ];
104 -> 12;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 105 [arrowhead = none ];
105 -> 13;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 106 [arrowhead = none ];
106 -> 74;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 107 [arrowhead = none ];
107 -> 75;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 108 [arrowhead = none ];
108 -> 14;

109 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
12 -> 109 [arrowhead = none ];
109 -> 15;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 110 [arrowhead = none ];
110 -> 16;

111 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 111 [arrowhead = none ];
111 -> 24;

112 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nreplacements:\nX52 -> T45\nX53 -> T46", fontsize=14, style = filled, fillcolor = violet];
15 -> 112 [arrowhead = none ];
112 -> 25;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 113 [arrowhead = none ];
113 -> 17;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 114 [arrowhead = none ];
114 -> 18;

115 [label="EVAL with clause\nless(0, s(X67)).\nand substitution\nT22 -> 0\nX67 -> T33\nT23 -> s(T33)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 115 [arrowhead = none ];
115 -> 19;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 116 [arrowhead = none ];
116 -> 20;

117 [label="EVAL with clause\nless(s(X73), s(X74)) :- less(X73, X74).\nand substitution\nX73 -> T38\nT22 -> s(T38)\nX74 -> T39\nT23 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 117 [arrowhead = none ];
117 -> 22;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 118 [arrowhead = none ];
118 -> 23;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 119 [arrowhead = none ];
119 -> 21;

120 [label="INSTANCE with matching:\nT22 -> T38\nT23 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 120 [arrowhead = none , style=dashed];
120 -> 14[style=dashed];

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 121 [arrowhead = none ];
121 -> 26;

122 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 122 [arrowhead = none ];
122 -> 40;

123 [label="SPLIT 2\nnew knowledge:\nT95 is ground\nreplacements:\nX14 -> T95", fontsize=14, style = filled, fillcolor = violet];
25 -> 123 [arrowhead = none ];
123 -> 41;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 124 [arrowhead = none ];
124 -> 27;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 125 [arrowhead = none ];
125 -> 28;

126 [label="EVAL with clause\npart(X121, .(X122, X123), .(X122, X124), X125) :- ','(less(X121, X122), part(X121, X123, X124, X125)).\nand substitution\nT22 -> T62\nX121 -> T62\nX122 -> T63\nX123 -> T64\nT24 -> .(T63, T64)\nX124 -> X126\nX52 -> .(T63, X126)\nX53 -> X127\nX125 -> X127", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 126 [arrowhead = none ];
126 -> 29;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 127 [arrowhead = none ];
127 -> 30;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 128 [arrowhead = none ];
128 -> 33;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 129 [arrowhead = none ];
129 -> 34;

130 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 130 [arrowhead = none ];
130 -> 31;

131 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
29 -> 131 [arrowhead = none ];
131 -> 32;

132 [label="INSTANCE with matching:\nT22 -> T62\nT23 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 132 [arrowhead = none , style=dashed];
132 -> 14[style=dashed];

133 [label="INSTANCE with matching:\nT22 -> T62\nT24 -> T64\nX52 -> X126\nX53 -> X127", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 133 [arrowhead = none , style=dashed];
133 -> 24[style=dashed];

134 [label="EVAL with clause\npart(X172, .(X173, X174), X175, .(X173, X176)) :- part(X172, X174, X175, X176).\nand substitution\nT22 -> T84\nX172 -> T84\nX173 -> T85\nX174 -> T86\nT24 -> .(T85, T86)\nX52 -> X177\nX175 -> X177\nX176 -> X178\nX53 -> .(T85, X178)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 134 [arrowhead = none ];
134 -> 35;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 135 [arrowhead = none ];
135 -> 36;

136 [label="EVAL with clause\npart(X193, [], [], []).\nand substitution\nT22 -> T92\nX193 -> T92\nT24 -> []\nX52 -> []\nX53 -> []", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 136 [arrowhead = none ];
136 -> 37;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 137 [arrowhead = none ];
137 -> 38;

138 [label="INSTANCE with matching:\nT22 -> T84\nT24 -> T86\nX52 -> X177\nX53 -> X178", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 138 [arrowhead = none , style=dashed];
138 -> 24[style=dashed];

139 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 139 [arrowhead = none ];
139 -> 39;

140 [label="INSTANCE with matching:\nT1 -> .(T23, T45)\nT2 -> X14", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 140 [arrowhead = none , style=dashed];
140 -> 1[style=dashed];

141 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 141 [arrowhead = none ];
141 -> 42;

142 [label="SPLIT 2\nnew knowledge:\nT98 is ground\nreplacements:\nX15 -> T98", fontsize=14, style = filled, fillcolor = violet];
41 -> 142 [arrowhead = none ];
142 -> 43;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 143 [arrowhead = none ];
143 -> 44;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 144 [arrowhead = none ];
144 -> 66;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 145 [arrowhead = none ];
145 -> 45;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 146 [arrowhead = none ];
146 -> 46;

147 [label="EVAL with clause\nqs([], []).\nand substitution\nT46 -> []\nX15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 147 [arrowhead = none ];
147 -> 47;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 148 [arrowhead = none ];
148 -> 48;

149 [label="EVAL with clause\nqs(.(X223, X224), X225) :- ','(part(X223, X224, X226, X227), ','(qs(X226, X228), ','(qs(X227, X229), app(X228, .(X223, X229), X225)))).\nand substitution\nX223 -> T103\nX224 -> T104\nT46 -> .(T103, T104)\nX15 -> X230\nX225 -> X230", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 149 [arrowhead = none ];
149 -> 50;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 150 [arrowhead = none ];
150 -> 51;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 151 [arrowhead = none ];
151 -> 49;

152 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 152 [arrowhead = none ];
152 -> 52;

153 [label="SPLIT 2\nnew knowledge:\nT108 is ground\nT109 is ground\nreplacements:\nX226 -> T108\nX227 -> T109", fontsize=14, style = filled, fillcolor = violet];
50 -> 153 [arrowhead = none ];
153 -> 53;

154 [label="INSTANCE with matching:\nT22 -> T103\nT24 -> T104\nX52 -> X226\nX53 -> X227", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 154 [arrowhead = none , style=dashed];
154 -> 24[style=dashed];

155 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 155 [arrowhead = none ];
155 -> 54;

156 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nreplacements:\nX228 -> T113", fontsize=14, style = filled, fillcolor = violet];
53 -> 156 [arrowhead = none ];
156 -> 55;

157 [label="INSTANCE with matching:\nT46 -> T108\nX15 -> X228", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 157 [arrowhead = none , style=dashed];
157 -> 42[style=dashed];

158 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
55 -> 158 [arrowhead = none ];
158 -> 56;

159 [label="SPLIT 2\nnew knowledge:\nT114 is ground\nreplacements:\nX229 -> T114", fontsize=14, style = filled, fillcolor = violet];
55 -> 159 [arrowhead = none ];
159 -> 57;

160 [label="INSTANCE with matching:\nT46 -> T109\nX15 -> X229", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 160 [arrowhead = none , style=dashed];
160 -> 42[style=dashed];

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 161 [arrowhead = none ];
161 -> 58;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 162 [arrowhead = none ];
162 -> 59;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 163 [arrowhead = none ];
163 -> 60;

164 [label="EVAL with clause\napp([], X254, X254).\nand substitution\nT113 -> []\nT103 -> T127\nT114 -> T128\nX254 -> .(T127, T128)\nX230 -> .(T127, T128)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 164 [arrowhead = none ];
164 -> 61;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 165 [arrowhead = none ];
165 -> 62;

166 [label="EVAL with clause\napp(.(X266, X267), X268, .(X266, X269)) :- app(X267, X268, X269).\nand substitution\nX266 -> T137\nX267 -> T138\nT113 -> .(T137, T138)\nT103 -> T139\nT114 -> T140\nX268 -> .(T139, T140)\nX269 -> X270\nX230 -> .(T137, X270)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 166 [arrowhead = none ];
166 -> 64;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 167 [arrowhead = none ];
167 -> 65;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 168 [arrowhead = none ];
168 -> 63;

169 [label="INSTANCE with matching:\nT113 -> T138\nT103 -> T139\nT114 -> T140\nX230 -> X270", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 169 [arrowhead = none , style=dashed];
169 -> 57[style=dashed];

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 170 [arrowhead = none ];
170 -> 67;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 171 [arrowhead = none ];
171 -> 68;

172 [label="EVAL with clause\napp([], X283, X283).\nand substitution\nT95 -> []\nT22 -> T157\nT98 -> T158\nX283 -> .(T157, T158)\nT9 -> .(T157, T158)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 172 [arrowhead = none ];
172 -> 69;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 173 [arrowhead = none ];
173 -> 70;

174 [label="EVAL with clause\napp(.(X293, X294), X295, .(X293, X296)) :- app(X294, X295, X296).\nand substitution\nX293 -> T169\nX294 -> T170\nT95 -> .(T169, T170)\nT22 -> T171\nT98 -> T172\nX295 -> .(T171, T172)\nX296 -> T174\nT9 -> .(T169, T174)\nT173 -> T174", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 174 [arrowhead = none ];
174 -> 72;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 175 [arrowhead = none ];
175 -> 73;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 176 [arrowhead = none ];
176 -> 71;

177 [label="INSTANCE with matching:\nT95 -> T170\nT22 -> T171\nT98 -> T172\nT9 -> T174", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 177 [arrowhead = none , style=dashed];
177 -> 43[style=dashed];

178 [label="EVAL with clause\npart(X331, .(X332, X333), X334, .(X332, X335)) :- part(X331, X333, X334, X335).\nand substitution\nT6 -> T191\nX331 -> T191\nX332 -> T192\nX333 -> T193\nT7 -> .(T192, T193)\nX12 -> X336\nX334 -> X336\nX335 -> X337\nX13 -> .(T192, X337)", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 178 [arrowhead = none ];
178 -> 76;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 179 [arrowhead = none ];
179 -> 77;

180 [label="EVAL with clause\npart(X367, [], [], []).\nand substitution\nT6 -> T207\nX367 -> T207\nT7 -> []\nX12 -> []\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 180 [arrowhead = none ];
180 -> 82;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 181 [arrowhead = none ];
181 -> 83;

182 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
76 -> 182 [arrowhead = none ];
182 -> 78;

183 [label="SPLIT 2\nnew knowledge:\nT197 is ground\nT198 is ground\nreplacements:\nX336 -> T197\nX337 -> T198", fontsize=14, style = filled, fillcolor = violet];
76 -> 183 [arrowhead = none ];
183 -> 79;

184 [label="INSTANCE with matching:\nT22 -> T191\nT24 -> T193\nX52 -> X336\nX53 -> X337", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 184 [arrowhead = none , style=dashed];
184 -> 24[style=dashed];

185 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
79 -> 185 [arrowhead = none ];
185 -> 80;

186 [label="SPLIT 2\nnew knowledge:\nT202 is ground\nreplacements:\nX14 -> T202", fontsize=14, style = filled, fillcolor = violet];
79 -> 186 [arrowhead = none ];
186 -> 81;

187 [label="INSTANCE with matching:\nT46 -> T197\nX15 -> X14", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 187 [arrowhead = none , style=dashed];
187 -> 42[style=dashed];

188 [label="INSTANCE with matching:\nT46 -> .(T192, T198)\nT95 -> T202\nT22 -> T191", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 188 [arrowhead = none , style=dashed];
188 -> 41[style=dashed];

189 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
82 -> 189 [arrowhead = none ];
189 -> 84;

190 [label="SPLIT 2\nnew knowledge:\nT210 is ground\nreplacements:\nX14 -> T210", fontsize=14, style = filled, fillcolor = violet];
82 -> 190 [arrowhead = none ];
190 -> 85;

191 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 191 [arrowhead = none ];
191 -> 86;

192 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 192 [arrowhead = none ];
192 -> 92;

193 [label="SPLIT 2\nnew knowledge:\nT213 is ground\nreplacements:\nX15 -> T213", fontsize=14, style = filled, fillcolor = violet];
85 -> 193 [arrowhead = none ];
193 -> 93;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 194 [arrowhead = none ];
194 -> 87;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 195 [arrowhead = none ];
195 -> 88;

196 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX14 -> []", fontsize=14, style = filled, fillcolor = white];
87 -> 196 [arrowhead = none ];
196 -> 89;

197 [label="BACKTRACK\nfor clause: qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
88 -> 197 [arrowhead = none ];
197 -> 91;

198 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 198 [arrowhead = none ];
198 -> 90;

199 [label="INSTANCE with matching:\nX14 -> X15", fontsize=14, style = filled, fillcolor = lightgrey];
92 -> 199 [arrowhead = none , style=dashed];
199 -> 84[style=dashed];

200 [label="INSTANCE with matching:\nT95 -> T210\nT22 -> T207\nT98 -> T213", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 200 [arrowhead = none , style=dashed];
200 -> 43[style=dashed];

}

(2) Obligation:

Clauses:

less14(0, s(T33)).
less14(s(T38), s(T39)) :- less14(T38, T39).
part24(T62, .(T63, T64), .(T63, X126), X127) :- less14(T62, T63).
part24(T62, .(T63, T64), .(T63, X126), X127) :- ','(less14(T62, T63), part24(T62, T64, X126, X127)).
part24(T84, .(T85, T86), X177, .(T85, X178)) :- part24(T84, T86, X177, X178).
part24(T92, [], [], []).
qs42([], []).
qs42(.(T103, T104), X230) :- part24(T103, T104, X226, X227).
qs42(.(T103, T104), X230) :- ','(part24(T103, T104, T108, T109), qs42(T108, X228)).
qs42(.(T103, T104), X230) :- ','(part24(T103, T104, T108, T109), ','(qs42(T108, T113), qs42(T109, X229))).
qs42(.(T103, T104), X230) :- ','(part24(T103, T104, T108, T109), ','(qs42(T108, T113), ','(qs42(T109, T114), app57(T113, T103, T114, X230)))).
app57([], T127, T128, .(T127, T128)).
app57(.(T137, T138), T139, T140, .(T137, X270)) :- app57(T138, T139, T140, X270).
app43([], T157, T158, .(T157, T158)).
app43(.(T169, T170), T171, T172, .(T169, T174)) :- app43(T170, T171, T172, T174).
p41(T46, X15, T95, T22, T9) :- qs42(T46, X15).
p41(T46, T98, T95, T22, T9) :- ','(qs42(T46, T98), app43(T95, T22, T98, T9)).
qs84([]).
qs1([], []).
qs1(.(T22, .(T23, T24)), T9) :- less14(T22, T23).
qs1(.(T22, .(T23, T24)), T9) :- ','(less14(T22, T23), part24(T22, T24, X52, X53)).
qs1(.(T22, .(T23, T24)), T9) :- ','(less14(T22, T23), ','(part24(T22, T24, T45, T46), qs1(.(T23, T45), X14))).
qs1(.(T22, .(T23, T24)), T9) :- ','(less14(T22, T23), ','(part24(T22, T24, T45, T46), ','(qs1(.(T23, T45), T95), p41(T46, X15, T95, T22, T9)))).
qs1(.(T191, .(T192, T193)), T9) :- part24(T191, T193, X336, X337).
qs1(.(T191, .(T192, T193)), T9) :- ','(part24(T191, T193, T197, T198), qs42(T197, X14)).
qs1(.(T191, .(T192, T193)), T9) :- ','(part24(T191, T193, T197, T198), ','(qs42(T197, T202), p41(.(T192, T198), X15, T202, T191, T9))).
qs1(.(T207, []), T9) :- qs84(X14).
qs1(.(T207, []), T9) :- ','(qs84(T210), qs84(X15)).
qs1(.(T207, []), T9) :- ','(qs84(T210), ','(qs84(T213), app43(T210, T207, T213, T9))).

Queries:

qs1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (b,f) (f,f)
less14_in: (b,b) (f,f) (b,f)
part24_in: (b,b,f,f) (b,f,f,f) (f,f,f,f)
qs42_in: (f,f)
app57_in: (f,f,f,f)
p41_in: (f,f,f,f,f) (f,f,f,b,f)
app43_in: (f,f,f,f) (b,f,b,f) (f,b,f,f) (b,b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U17_GA(T22, T23, T24, T9, less14_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → LESS14_IN_GG(T22, T23)
LESS14_IN_GG(s(T38), s(T39)) → U1_GG(T38, T39, less14_in_gg(T38, T39))
LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → PART24_IN_GGAA(T22, T24, X52, X53)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GG(T62, T63)
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_GGAA(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GGAA(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → LESS14_IN_AA(T22, T23)
LESS14_IN_AA(s(T38), s(T39)) → U1_AA(T38, T39, less14_in_aa(T38, T39))
LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → PART24_IN_GAAA(T22, T24, X52, X53)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GA(T62, T63)
LESS14_IN_GA(s(T38), s(T39)) → U1_GA(T38, T39, less14_in_ga(T38, T39))
LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_GAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GAAA(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U23_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
QS1_IN_AA(.(T191, .(T192, T193)), T9) → PART24_IN_AAAA(T191, T193, X336, X337)
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_AAAA(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_AA(T62, T63)
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_AAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_AAAA(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U24_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_AA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
QS42_IN_AA(.(T103, T104), X230) → U5_AA(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
QS42_IN_AA(.(T103, T104), X230) → PART24_IN_AAAA(T103, T104, X226, X227)
QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_AA(T103, T104, X230, qs42_in_aa(T108, X228))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_AA(T103, T104, X230, qs42_in_aa(T109, X229))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_AA(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_AA(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → APP57_IN_AAAA(T113, T103, T114, X230)
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → U12_AAAA(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_AA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_AA(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAAA(.(T192, T198), X15, T202, T191, T9)
P41_IN_AAAAA(T46, X15, T95, T22, T9) → U14_AAAAA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAAA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAAA(T46, T98, T95, T22, T9) → U15_AAAAA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAAA(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AAAA(T95, T22, T98, T9)
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AAAA(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)
QS1_IN_AA(.(T207, []), T9) → U28_AA(T207, T9, qs84_in_a(X14))
QS1_IN_AA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_AA(.(T207, []), T9) → U29_AA(T207, T9, qs84_in_a(T210))
U29_AA(T207, T9, qs84_out_a(T210)) → U30_AA(T207, T9, qs84_in_a(X15))
U29_AA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_AA(T207, T9, qs84_out_a(T210)) → U31_AA(T207, T9, T210, qs84_in_a(T213))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → U32_AA(T207, T9, app43_in_gaga(T210, T207, T213, T9))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GAGA(T210, T207, T213, T9)
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GAGA(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_AA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_AA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
P41_IN_AAAGA(T46, X15, T95, T22, T9) → U14_AAAGA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAGA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAGA(T46, T98, T95, T22, T9) → U15_AAAGA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAGA(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AGAA(T95, T22, T98, T9)
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AGAA(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_GA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_GA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U23_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
QS1_IN_GA(.(T191, .(T192, T193)), T9) → PART24_IN_GGAA(T191, T193, X336, X337)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U24_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_GA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_GA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_GA(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAGA(.(T192, T198), X15, T202, T191, T9)
QS1_IN_GA(.(T207, []), T9) → U28_GA(T207, T9, qs84_in_a(X14))
QS1_IN_GA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_GA(.(T207, []), T9) → U29_GA(T207, T9, qs84_in_a(T210))
U29_GA(T207, T9, qs84_out_a(T210)) → U30_GA(T207, T9, qs84_in_a(X15))
U29_GA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_GA(T207, T9, qs84_out_a(T210)) → U31_GA(T207, T9, T210, qs84_in_a(T213))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → U32_GA(T207, T9, app43_in_ggga(T210, T207, T213, T9))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GGGA(T210, T207, T213, T9)
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GGGA(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x3, x5)
LESS14_IN_GG(x1, x2)  =  LESS14_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x6)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(x5)
U23_AA(x1, x2, x3, x4, x5)  =  U23_AA(x5)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5, x6)  =  U2_AAAA(x6)
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA
U12_AAAA(x1, x2, x3, x4, x5, x6)  =  U12_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5, x6)  =  U26_AA(x6)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
P41_IN_AAAAA(x1, x2, x3, x4, x5)  =  P41_IN_AAAAA
U14_AAAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAAA(x6)
U15_AAAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAAA(x6)
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA
U13_AAAA(x1, x2, x3, x4, x5, x6)  =  U13_AAAA(x6)
U28_AA(x1, x2, x3)  =  U28_AA(x3)
QS84_IN_A(x1)  =  QS84_IN_A
U29_AA(x1, x2, x3)  =  U29_AA(x3)
U30_AA(x1, x2, x3)  =  U30_AA(x3)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x3, x4)
U32_AA(x1, x2, x3)  =  U32_AA(x3)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)
U13_GAGA(x1, x2, x3, x4, x5, x6)  =  U13_GAGA(x6)
U21_AA(x1, x2, x3, x4, x5, x6)  =  U21_AA(x1, x6)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
P41_IN_AAAGA(x1, x2, x3, x4, x5)  =  P41_IN_AAAGA(x4)
U14_AAAGA(x1, x2, x3, x4, x5, x6)  =  U14_AAAGA(x6)
U15_AAAGA(x1, x2, x3, x4, x5, x6)  =  U15_AAAGA(x4, x6)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)
U13_AGAA(x1, x2, x3, x4, x5, x6)  =  U13_AGAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x1, x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x5)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x5)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
U26_GA(x1, x2, x3, x4, x5, x6)  =  U26_GA(x1, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x5)
U28_GA(x1, x2, x3)  =  U28_GA(x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x3)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x1, x3, x4)
U32_GA(x1, x2, x3)  =  U32_GA(x3)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4, x5, x6)  =  U13_GGGA(x1, x6)

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

(6) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U17_GA(T22, T23, T24, T9, less14_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → LESS14_IN_GG(T22, T23)
LESS14_IN_GG(s(T38), s(T39)) → U1_GG(T38, T39, less14_in_gg(T38, T39))
LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → PART24_IN_GGAA(T22, T24, X52, X53)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GG(T62, T63)
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_GGAA(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GGAA(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → LESS14_IN_AA(T22, T23)
LESS14_IN_AA(s(T38), s(T39)) → U1_AA(T38, T39, less14_in_aa(T38, T39))
LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → PART24_IN_GAAA(T22, T24, X52, X53)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GA(T62, T63)
LESS14_IN_GA(s(T38), s(T39)) → U1_GA(T38, T39, less14_in_ga(T38, T39))
LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_GAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GAAA(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U23_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
QS1_IN_AA(.(T191, .(T192, T193)), T9) → PART24_IN_AAAA(T191, T193, X336, X337)
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_AAAA(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_AA(T62, T63)
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_AAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_AAAA(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U24_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_AA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
QS42_IN_AA(.(T103, T104), X230) → U5_AA(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
QS42_IN_AA(.(T103, T104), X230) → PART24_IN_AAAA(T103, T104, X226, X227)
QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_AA(T103, T104, X230, qs42_in_aa(T108, X228))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_AA(T103, T104, X230, qs42_in_aa(T109, X229))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_AA(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_AA(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → APP57_IN_AAAA(T113, T103, T114, X230)
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → U12_AAAA(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_AA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_AA(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAAA(.(T192, T198), X15, T202, T191, T9)
P41_IN_AAAAA(T46, X15, T95, T22, T9) → U14_AAAAA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAAA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAAA(T46, T98, T95, T22, T9) → U15_AAAAA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAAA(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AAAA(T95, T22, T98, T9)
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AAAA(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)
QS1_IN_AA(.(T207, []), T9) → U28_AA(T207, T9, qs84_in_a(X14))
QS1_IN_AA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_AA(.(T207, []), T9) → U29_AA(T207, T9, qs84_in_a(T210))
U29_AA(T207, T9, qs84_out_a(T210)) → U30_AA(T207, T9, qs84_in_a(X15))
U29_AA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_AA(T207, T9, qs84_out_a(T210)) → U31_AA(T207, T9, T210, qs84_in_a(T213))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → U32_AA(T207, T9, app43_in_gaga(T210, T207, T213, T9))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GAGA(T210, T207, T213, T9)
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GAGA(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_AA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_AA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
P41_IN_AAAGA(T46, X15, T95, T22, T9) → U14_AAAGA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAGA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAGA(T46, T98, T95, T22, T9) → U15_AAAGA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAGA(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AGAA(T95, T22, T98, T9)
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AGAA(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_GA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_GA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U23_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
QS1_IN_GA(.(T191, .(T192, T193)), T9) → PART24_IN_GGAA(T191, T193, X336, X337)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U24_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_GA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_GA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_GA(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAGA(.(T192, T198), X15, T202, T191, T9)
QS1_IN_GA(.(T207, []), T9) → U28_GA(T207, T9, qs84_in_a(X14))
QS1_IN_GA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_GA(.(T207, []), T9) → U29_GA(T207, T9, qs84_in_a(T210))
U29_GA(T207, T9, qs84_out_a(T210)) → U30_GA(T207, T9, qs84_in_a(X15))
U29_GA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_GA(T207, T9, qs84_out_a(T210)) → U31_GA(T207, T9, T210, qs84_in_a(T213))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → U32_GA(T207, T9, app43_in_ggga(T210, T207, T213, T9))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GGGA(T210, T207, T213, T9)
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GGGA(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x3, x5)
LESS14_IN_GG(x1, x2)  =  LESS14_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x6)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(x5)
U23_AA(x1, x2, x3, x4, x5)  =  U23_AA(x5)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5, x6)  =  U2_AAAA(x6)
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA
U12_AAAA(x1, x2, x3, x4, x5, x6)  =  U12_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5, x6)  =  U26_AA(x6)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
P41_IN_AAAAA(x1, x2, x3, x4, x5)  =  P41_IN_AAAAA
U14_AAAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAAA(x6)
U15_AAAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAAA(x6)
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA
U13_AAAA(x1, x2, x3, x4, x5, x6)  =  U13_AAAA(x6)
U28_AA(x1, x2, x3)  =  U28_AA(x3)
QS84_IN_A(x1)  =  QS84_IN_A
U29_AA(x1, x2, x3)  =  U29_AA(x3)
U30_AA(x1, x2, x3)  =  U30_AA(x3)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x3, x4)
U32_AA(x1, x2, x3)  =  U32_AA(x3)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)
U13_GAGA(x1, x2, x3, x4, x5, x6)  =  U13_GAGA(x6)
U21_AA(x1, x2, x3, x4, x5, x6)  =  U21_AA(x1, x6)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
P41_IN_AAAGA(x1, x2, x3, x4, x5)  =  P41_IN_AAAGA(x4)
U14_AAAGA(x1, x2, x3, x4, x5, x6)  =  U14_AAAGA(x6)
U15_AAAGA(x1, x2, x3, x4, x5, x6)  =  U15_AAAGA(x4, x6)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)
U13_AGAA(x1, x2, x3, x4, x5, x6)  =  U13_AGAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x1, x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x5)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x5)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
U26_GA(x1, x2, x3, x4, x5, x6)  =  U26_GA(x1, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x5)
U28_GA(x1, x2, x3)  =  U28_GA(x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x3)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x1, x3, x4)
U32_GA(x1, x2, x3)  =  U32_GA(x3)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4, x5, x6)  =  U13_GGGA(x1, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 13 SCCs with 85 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_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:

APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_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:

APP43_IN_GGGA(.(T169, T170), T171, T172) → APP43_IN_GGGA(T170, T171, T172)

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:

• APP43_IN_GGGA(.(T169, T170), T171, T172) → APP43_IN_GGGA(T170, T171, T172)
  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:

APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP43_IN_AGAA(T171) → APP43_IN_AGAA(T171)

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

(21) NonTerminationProof (EQUIVALENT transformation)

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

s = APP43_IN_AGAA(T171) evaluates to t =APP43_IN_AGAA(T171)

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP43_IN_AGAA(T171) to APP43_IN_AGAA(T171).

(22) NO

(23) Obligation:

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

APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)

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:

APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APP43_IN_GAGA(.(T169, T170), T172) → APP43_IN_GAGA(T170, T172)

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:

• APP43_IN_GAGA(.(T169, T170), T172) → APP43_IN_GAGA(T170, T172)
  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:

APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA

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:

APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA

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:

APP43_IN_AAAA → APP43_IN_AAAA

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

(35) NonTerminationProof (EQUIVALENT transformation)

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

s = APP43_IN_AAAA evaluates to t =APP43_IN_AAAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP43_IN_AAAA to APP43_IN_AAAA.

(36) NO

(37) Obligation:

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

APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA

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:

APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA

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:

APP57_IN_AAAA → APP57_IN_AAAA

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

(42) NonTerminationProof (EQUIVALENT transformation)

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

s = APP57_IN_AAAA evaluates to t =APP57_IN_AAAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP57_IN_AAAA to APP57_IN_AAAA.

(43) NO

(44) Obligation:

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

LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

LESS14_IN_GA(s(T38)) → LESS14_IN_GA(T38)

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

(49) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS14_IN_GA(s(T38)) → LESS14_IN_GA(T38)
  The graph contains the following edges 1 > 1

(50) YES

(51) Obligation:

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

U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)

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

(54) PiDPToQDPProof (SOUND transformation)

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

(55) Obligation:

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

U2_GAAA(T62, less14_out_ga) → PART24_IN_GAAA(T62)
PART24_IN_GAAA(T62) → U2_GAAA(T62, less14_in_ga(T62))
PART24_IN_GAAA(T84) → PART24_IN_GAAA(T84)

The TRS R consists of the following rules:

less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U1_ga(less14_out_ga) → less14_out_ga

The set Q consists of the following terms:

less14_in_ga(x0)
U1_ga(x0)

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

(56) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART24_IN_GAAA(T62) → U2_GAAA(T62, less14_in_ga(T62)) at position [1] we obtained the following new rules [LPAR04]:

PART24_IN_GAAA(0) → U2_GAAA(0, less14_out_ga)
PART24_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(less14_in_ga(x0)))

(57) Obligation:

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

U2_GAAA(T62, less14_out_ga) → PART24_IN_GAAA(T62)
PART24_IN_GAAA(T84) → PART24_IN_GAAA(T84)
PART24_IN_GAAA(0) → U2_GAAA(0, less14_out_ga)
PART24_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(less14_in_ga(x0)))

The TRS R consists of the following rules:

less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U1_ga(less14_out_ga) → less14_out_ga

The set Q consists of the following terms:

less14_in_ga(x0)
U1_ga(x0)

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

(58) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GAAA(T62, less14_out_ga) → PART24_IN_GAAA(T62) we obtained the following new rules [LPAR04]:

U2_GAAA(0, less14_out_ga) → PART24_IN_GAAA(0)
U2_GAAA(s(z0), less14_out_ga) → PART24_IN_GAAA(s(z0))

(59) Obligation:

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

PART24_IN_GAAA(T84) → PART24_IN_GAAA(T84)
PART24_IN_GAAA(0) → U2_GAAA(0, less14_out_ga)
PART24_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(less14_in_ga(x0)))
U2_GAAA(0, less14_out_ga) → PART24_IN_GAAA(0)
U2_GAAA(s(z0), less14_out_ga) → PART24_IN_GAAA(s(z0))

The TRS R consists of the following rules:

less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U1_ga(less14_out_ga) → less14_out_ga

The set Q consists of the following terms:

less14_in_ga(x0)
U1_ga(x0)

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

(60) NonTerminationProof (EQUIVALENT transformation)

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

s = PART24_IN_GAAA(T84) evaluates to t =PART24_IN_GAAA(T84)

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART24_IN_GAAA(T84) to PART24_IN_GAAA(T84).

(61) NO

(62) Obligation:

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

LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA

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

(63) UsableRulesProof (EQUIVALENT transformation)

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

(64) Obligation:

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

LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA

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

(65) PiDPToQDPProof (SOUND transformation)

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

(66) Obligation:

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

LESS14_IN_AA → LESS14_IN_AA

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

(67) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS14_IN_AA evaluates to t =LESS14_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS14_IN_AA to LESS14_IN_AA.

(68) NO

(69) Obligation:

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

PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA

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

(70) UsableRulesProof (EQUIVALENT transformation)

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

(71) Obligation:

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

PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA

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

(72) PiDPToQDPProof (SOUND transformation)

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

(73) Obligation:

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

PART24_IN_AAAA → PART24_IN_AAAA

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

(74) NonTerminationProof (EQUIVALENT transformation)

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

s = PART24_IN_AAAA evaluates to t =PART24_IN_AAAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART24_IN_AAAA to PART24_IN_AAAA.

(75) NO

(76) Obligation:

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

QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

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

QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)

The TRS R consists of the following rules:

part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)

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

(79) PiDPToQDPProof (SOUND transformation)

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

(80) Obligation:

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

QS42_IN_AA → U6_AA(part24_in_aaaa)
U6_AA(part24_out_aaaa) → QS42_IN_AA
U6_AA(part24_out_aaaa) → U8_AA(qs42_in_aa)
U8_AA(qs42_out_aa) → QS42_IN_AA

The TRS R consists of the following rules:

part24_in_aaaa → U2_aaaa(less14_in_aa)
part24_in_aaaa → U4_aaaa(part24_in_aaaa)
part24_in_aaaa → part24_out_aaaa
qs42_in_aa → qs42_out_aa
qs42_in_aa → U5_aa(part24_in_aaaa)
qs42_in_aa → U6_aa(part24_in_aaaa)
U2_aaaa(less14_out_aa(T62)) → part24_out_aaaa
U2_aaaa(less14_out_aa(T62)) → U3_aaaa(part24_in_gaaa(T62))
U4_aaaa(part24_out_aaaa) → part24_out_aaaa
U5_aa(part24_out_aaaa) → qs42_out_aa
U6_aa(part24_out_aaaa) → U7_aa(qs42_in_aa)
U6_aa(part24_out_aaaa) → U8_aa(qs42_in_aa)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U3_aaaa(part24_out_gaaa) → part24_out_aaaa
U7_aa(qs42_out_aa) → qs42_out_aa
U8_aa(qs42_out_aa) → U9_aa(qs42_in_aa)
U8_aa(qs42_out_aa) → U10_aa(qs42_in_aa)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa
U9_aa(qs42_out_aa) → qs42_out_aa
U10_aa(qs42_out_aa) → U11_aa(app57_in_aaaa)
U2_gaaa(T62, less14_out_ga) → part24_out_gaaa
U2_gaaa(T62, less14_out_ga) → U3_gaaa(part24_in_gaaa(T62))
U4_gaaa(part24_out_gaaa) → part24_out_gaaa
U11_aa(app57_out_aaaa) → qs42_out_aa
less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U3_gaaa(part24_out_gaaa) → part24_out_gaaa
app57_in_aaaa → app57_out_aaaa
app57_in_aaaa → U12_aaaa(app57_in_aaaa)
U1_ga(less14_out_ga) → less14_out_ga
U12_aaaa(app57_out_aaaa) → app57_out_aaaa

The set Q consists of the following terms:

part24_in_aaaa
qs42_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U5_aa(x0)
U6_aa(x0)
less14_in_aa
U3_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
U1_aa(x0)
part24_in_gaaa(x0)
U9_aa(x0)
U10_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0)
U11_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0)
app57_in_aaaa
U1_ga(x0)
U12_aaaa(x0)

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

(81) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS42_IN_AA → U6_AA(part24_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS42_IN_AA → U6_AA(U2_aaaa(less14_in_aa))
QS42_IN_AA → U6_AA(U4_aaaa(part24_in_aaaa))
QS42_IN_AA → U6_AA(part24_out_aaaa)

(82) Obligation:

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

U6_AA(part24_out_aaaa) → QS42_IN_AA
U6_AA(part24_out_aaaa) → U8_AA(qs42_in_aa)
U8_AA(qs42_out_aa) → QS42_IN_AA
QS42_IN_AA → U6_AA(U2_aaaa(less14_in_aa))
QS42_IN_AA → U6_AA(U4_aaaa(part24_in_aaaa))
QS42_IN_AA → U6_AA(part24_out_aaaa)

The TRS R consists of the following rules:

part24_in_aaaa → U2_aaaa(less14_in_aa)
part24_in_aaaa → U4_aaaa(part24_in_aaaa)
part24_in_aaaa → part24_out_aaaa
qs42_in_aa → qs42_out_aa
qs42_in_aa → U5_aa(part24_in_aaaa)
qs42_in_aa → U6_aa(part24_in_aaaa)
U2_aaaa(less14_out_aa(T62)) → part24_out_aaaa
U2_aaaa(less14_out_aa(T62)) → U3_aaaa(part24_in_gaaa(T62))
U4_aaaa(part24_out_aaaa) → part24_out_aaaa
U5_aa(part24_out_aaaa) → qs42_out_aa
U6_aa(part24_out_aaaa) → U7_aa(qs42_in_aa)
U6_aa(part24_out_aaaa) → U8_aa(qs42_in_aa)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U3_aaaa(part24_out_gaaa) → part24_out_aaaa
U7_aa(qs42_out_aa) → qs42_out_aa
U8_aa(qs42_out_aa) → U9_aa(qs42_in_aa)
U8_aa(qs42_out_aa) → U10_aa(qs42_in_aa)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa
U9_aa(qs42_out_aa) → qs42_out_aa
U10_aa(qs42_out_aa) → U11_aa(app57_in_aaaa)
U2_gaaa(T62, less14_out_ga) → part24_out_gaaa
U2_gaaa(T62, less14_out_ga) → U3_gaaa(part24_in_gaaa(T62))
U4_gaaa(part24_out_gaaa) → part24_out_gaaa
U11_aa(app57_out_aaaa) → qs42_out_aa
less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U3_gaaa(part24_out_gaaa) → part24_out_gaaa
app57_in_aaaa → app57_out_aaaa
app57_in_aaaa → U12_aaaa(app57_in_aaaa)
U1_ga(less14_out_ga) → less14_out_ga
U12_aaaa(app57_out_aaaa) → app57_out_aaaa

The set Q consists of the following terms:

part24_in_aaaa
qs42_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U5_aa(x0)
U6_aa(x0)
less14_in_aa
U3_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
U1_aa(x0)
part24_in_gaaa(x0)
U9_aa(x0)
U10_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0)
U11_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0)
app57_in_aaaa
U1_ga(x0)
U12_aaaa(x0)

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

(83) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U6_AA(part24_out_aaaa) → U8_AA(qs42_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U6_AA(part24_out_aaaa) → U8_AA(qs42_out_aa)
U6_AA(part24_out_aaaa) → U8_AA(U5_aa(part24_in_aaaa))
U6_AA(part24_out_aaaa) → U8_AA(U6_aa(part24_in_aaaa))

(84) Obligation:

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

U6_AA(part24_out_aaaa) → QS42_IN_AA
U8_AA(qs42_out_aa) → QS42_IN_AA
QS42_IN_AA → U6_AA(U2_aaaa(less14_in_aa))
QS42_IN_AA → U6_AA(U4_aaaa(part24_in_aaaa))
QS42_IN_AA → U6_AA(part24_out_aaaa)
U6_AA(part24_out_aaaa) → U8_AA(qs42_out_aa)
U6_AA(part24_out_aaaa) → U8_AA(U5_aa(part24_in_aaaa))
U6_AA(part24_out_aaaa) → U8_AA(U6_aa(part24_in_aaaa))

The TRS R consists of the following rules:

part24_in_aaaa → U2_aaaa(less14_in_aa)
part24_in_aaaa → U4_aaaa(part24_in_aaaa)
part24_in_aaaa → part24_out_aaaa
qs42_in_aa → qs42_out_aa
qs42_in_aa → U5_aa(part24_in_aaaa)
qs42_in_aa → U6_aa(part24_in_aaaa)
U2_aaaa(less14_out_aa(T62)) → part24_out_aaaa
U2_aaaa(less14_out_aa(T62)) → U3_aaaa(part24_in_gaaa(T62))
U4_aaaa(part24_out_aaaa) → part24_out_aaaa
U5_aa(part24_out_aaaa) → qs42_out_aa
U6_aa(part24_out_aaaa) → U7_aa(qs42_in_aa)
U6_aa(part24_out_aaaa) → U8_aa(qs42_in_aa)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U3_aaaa(part24_out_gaaa) → part24_out_aaaa
U7_aa(qs42_out_aa) → qs42_out_aa
U8_aa(qs42_out_aa) → U9_aa(qs42_in_aa)
U8_aa(qs42_out_aa) → U10_aa(qs42_in_aa)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa
U9_aa(qs42_out_aa) → qs42_out_aa
U10_aa(qs42_out_aa) → U11_aa(app57_in_aaaa)
U2_gaaa(T62, less14_out_ga) → part24_out_gaaa
U2_gaaa(T62, less14_out_ga) → U3_gaaa(part24_in_gaaa(T62))
U4_gaaa(part24_out_gaaa) → part24_out_gaaa
U11_aa(app57_out_aaaa) → qs42_out_aa
less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U3_gaaa(part24_out_gaaa) → part24_out_gaaa
app57_in_aaaa → app57_out_aaaa
app57_in_aaaa → U12_aaaa(app57_in_aaaa)
U1_ga(less14_out_ga) → less14_out_ga
U12_aaaa(app57_out_aaaa) → app57_out_aaaa

The set Q consists of the following terms:

part24_in_aaaa
qs42_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U5_aa(x0)
U6_aa(x0)
less14_in_aa
U3_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
U1_aa(x0)
part24_in_gaaa(x0)
U9_aa(x0)
U10_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0)
U11_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0)
app57_in_aaaa
U1_ga(x0)
U12_aaaa(x0)

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

(85) NonTerminationProof (EQUIVALENT transformation)

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

s = QS42_IN_AA evaluates to t =QS42_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS42_IN_AA → U6_AA(part24_out_aaaa)
with rule QS42_IN_AA → U6_AA(part24_out_aaaa) at position [] and matcher [ ]

U6_AA(part24_out_aaaa) → QS42_IN_AA
with rule U6_AA(part24_out_aaaa) → QS42_IN_AA

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


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

(86) NO

(87) Obligation:

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

U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)

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

(88) UsableRulesProof (EQUIVALENT transformation)

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

(89) Obligation:

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

U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))

The TRS R consists of the following rules:

part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)

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

(90) PiDPToQDPProof (SOUND transformation)

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

(91) Obligation:

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

U17_AA(less14_out_aa(T22)) → U19_AA(T22, part24_in_gaaa(T22))
U19_AA(T22, part24_out_gaaa) → QS1_IN_AA
QS1_IN_AA → U17_AA(less14_in_aa)

The TRS R consists of the following rules:

part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U2_gaaa(T62, less14_out_ga) → part24_out_gaaa
U2_gaaa(T62, less14_out_ga) → U3_gaaa(part24_in_gaaa(T62))
U4_gaaa(part24_out_gaaa) → part24_out_gaaa
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U3_gaaa(part24_out_gaaa) → part24_out_gaaa
U1_ga(less14_out_ga) → less14_out_ga

The set Q consists of the following terms:

part24_in_gaaa(x0)
less14_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0)
U1_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0)
U1_ga(x0)

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

(92) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U17_AA(less14_out_aa(T22)) → U19_AA(T22, part24_in_gaaa(T22)) at position [1] we obtained the following new rules [LPAR04]:

U17_AA(less14_out_aa(x0)) → U19_AA(x0, U2_gaaa(x0, less14_in_ga(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U4_gaaa(part24_in_gaaa(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa)

(93) Obligation:

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

U19_AA(T22, part24_out_gaaa) → QS1_IN_AA
QS1_IN_AA → U17_AA(less14_in_aa)
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U2_gaaa(x0, less14_in_ga(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U4_gaaa(part24_in_gaaa(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa)

The TRS R consists of the following rules:

part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U2_gaaa(T62, less14_out_ga) → part24_out_gaaa
U2_gaaa(T62, less14_out_ga) → U3_gaaa(part24_in_gaaa(T62))
U4_gaaa(part24_out_gaaa) → part24_out_gaaa
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U3_gaaa(part24_out_gaaa) → part24_out_gaaa
U1_ga(less14_out_ga) → less14_out_ga

The set Q consists of the following terms:

part24_in_gaaa(x0)
less14_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0)
U1_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0)
U1_ga(x0)

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

(94) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS1_IN_AA → U17_AA(less14_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QS1_IN_AA → U17_AA(less14_out_aa(0))
QS1_IN_AA → U17_AA(U1_aa(less14_in_aa))

(95) Obligation:

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

U19_AA(T22, part24_out_gaaa) → QS1_IN_AA
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U2_gaaa(x0, less14_in_ga(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U4_gaaa(part24_in_gaaa(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa)
QS1_IN_AA → U17_AA(less14_out_aa(0))
QS1_IN_AA → U17_AA(U1_aa(less14_in_aa))

The TRS R consists of the following rules:

part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U2_gaaa(T62, less14_out_ga) → part24_out_gaaa
U2_gaaa(T62, less14_out_ga) → U3_gaaa(part24_in_gaaa(T62))
U4_gaaa(part24_out_gaaa) → part24_out_gaaa
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
less14_in_ga(0) → less14_out_ga
less14_in_ga(s(T38)) → U1_ga(less14_in_ga(T38))
U3_gaaa(part24_out_gaaa) → part24_out_gaaa
U1_ga(less14_out_ga) → less14_out_ga

The set Q consists of the following terms:

part24_in_gaaa(x0)
less14_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0)
U1_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0)
U1_ga(x0)

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

(96) NonTerminationProof (EQUIVALENT transformation)

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

s = QS1_IN_AA evaluates to t =QS1_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

QS1_IN_AA → U17_AA(less14_out_aa(0))
with rule QS1_IN_AA → U17_AA(less14_out_aa(0)) at position [] and matcher [ ]

U17_AA(less14_out_aa(0)) → U19_AA(0, part24_out_gaaa)
with rule U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa) at position [] and matcher [x0 / 0]

U19_AA(0, part24_out_gaaa) → QS1_IN_AA
with rule U19_AA(T22, part24_out_gaaa) → QS1_IN_AA

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


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

(97) NO

(98) Obligation:

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

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
LESS14_IN_GG(x1, x2)  =  LESS14_IN_GG(x1, x2)

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

(99) UsableRulesProof (EQUIVALENT transformation)

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

(100) Obligation:

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

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

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

(101) PiDPToQDPProof (EQUIVALENT transformation)

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

(102) Obligation:

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

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

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

(103) 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:

• LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
  The graph contains the following edges 1 > 1, 2 > 2

(104) YES

(105) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x6)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)

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

(106) UsableRulesProof (EQUIVALENT transformation)

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

(107) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)

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

(108) PiDPToQDPProof (SOUND transformation)

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

(109) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64)) → U2_GGAA(T62, T64, less14_in_gg(T62, T63))
U2_GGAA(T62, T64, less14_out_gg) → PART24_IN_GGAA(T62, T64)
PART24_IN_GGAA(T84, .(T85, T86)) → PART24_IN_GGAA(T84, T86)

The TRS R consists of the following rules:

less14_in_gg(0, s(T33)) → less14_out_gg
less14_in_gg(s(T38), s(T39)) → U1_gg(less14_in_gg(T38, T39))
U1_gg(less14_out_gg) → less14_out_gg

The set Q consists of the following terms:

less14_in_gg(x0, x1)
U1_gg(x0)

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

(110) 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:

• U2_GGAA(T62, T64, less14_out_gg) → PART24_IN_GGAA(T62, T64)
  The graph contains the following edges 1 >= 1, 2 >= 2

• PART24_IN_GGAA(T84, .(T85, T86)) → PART24_IN_GGAA(T84, T86)
  The graph contains the following edges 1 >= 1, 2 > 2

• PART24_IN_GGAA(T62, .(T63, T64)) → U2_GGAA(T62, T64, less14_in_gg(T62, T63))
  The graph contains the following edges 1 >= 1, 2 > 2

(111) YES

(112) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (b,f) (f,f)
less14_in: (b,b) (f,f) (b,f)
part24_in: (b,b,f,f) (b,f,f,f) (f,f,f,f)
qs42_in: (f,f)
app57_in: (f,f,f,f)
p41_in: (f,f,f,f,f) (f,f,f,b,f)
app43_in: (f,f,f,f) (b,f,b,f) (f,b,f,f) (b,b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(113) Obligation:

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

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)

(114) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U17_GA(T22, T23, T24, T9, less14_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → LESS14_IN_GG(T22, T23)
LESS14_IN_GG(s(T38), s(T39)) → U1_GG(T38, T39, less14_in_gg(T38, T39))
LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → PART24_IN_GGAA(T22, T24, X52, X53)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GG(T62, T63)
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_GGAA(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GGAA(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → LESS14_IN_AA(T22, T23)
LESS14_IN_AA(s(T38), s(T39)) → U1_AA(T38, T39, less14_in_aa(T38, T39))
LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → PART24_IN_GAAA(T22, T24, X52, X53)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GA(T62, T63)
LESS14_IN_GA(s(T38), s(T39)) → U1_GA(T38, T39, less14_in_ga(T38, T39))
LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_GAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GAAA(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U23_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
QS1_IN_AA(.(T191, .(T192, T193)), T9) → PART24_IN_AAAA(T191, T193, X336, X337)
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_AAAA(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_AA(T62, T63)
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_AAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_AAAA(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U24_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_AA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
QS42_IN_AA(.(T103, T104), X230) → U5_AA(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
QS42_IN_AA(.(T103, T104), X230) → PART24_IN_AAAA(T103, T104, X226, X227)
QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_AA(T103, T104, X230, qs42_in_aa(T108, X228))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_AA(T103, T104, X230, qs42_in_aa(T109, X229))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_AA(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_AA(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → APP57_IN_AAAA(T113, T103, T114, X230)
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → U12_AAAA(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_AA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_AA(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAAA(.(T192, T198), X15, T202, T191, T9)
P41_IN_AAAAA(T46, X15, T95, T22, T9) → U14_AAAAA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAAA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAAA(T46, T98, T95, T22, T9) → U15_AAAAA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAAA(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AAAA(T95, T22, T98, T9)
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AAAA(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)
QS1_IN_AA(.(T207, []), T9) → U28_AA(T207, T9, qs84_in_a(X14))
QS1_IN_AA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_AA(.(T207, []), T9) → U29_AA(T207, T9, qs84_in_a(T210))
U29_AA(T207, T9, qs84_out_a(T210)) → U30_AA(T207, T9, qs84_in_a(X15))
U29_AA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_AA(T207, T9, qs84_out_a(T210)) → U31_AA(T207, T9, T210, qs84_in_a(T213))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → U32_AA(T207, T9, app43_in_gaga(T210, T207, T213, T9))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GAGA(T210, T207, T213, T9)
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GAGA(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_AA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_AA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
P41_IN_AAAGA(T46, X15, T95, T22, T9) → U14_AAAGA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAGA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAGA(T46, T98, T95, T22, T9) → U15_AAAGA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAGA(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AGAA(T95, T22, T98, T9)
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AGAA(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_GA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_GA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U23_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
QS1_IN_GA(.(T191, .(T192, T193)), T9) → PART24_IN_GGAA(T191, T193, X336, X337)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U24_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_GA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_GA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_GA(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAGA(.(T192, T198), X15, T202, T191, T9)
QS1_IN_GA(.(T207, []), T9) → U28_GA(T207, T9, qs84_in_a(X14))
QS1_IN_GA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_GA(.(T207, []), T9) → U29_GA(T207, T9, qs84_in_a(T210))
U29_GA(T207, T9, qs84_out_a(T210)) → U30_GA(T207, T9, qs84_in_a(X15))
U29_GA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_GA(T207, T9, qs84_out_a(T210)) → U31_GA(T207, T9, T210, qs84_in_a(T213))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → U32_GA(T207, T9, app43_in_ggga(T210, T207, T213, T9))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GGGA(T210, T207, T213, T9)
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GGGA(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)
LESS14_IN_GG(x1, x2)  =  LESS14_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x3, x5)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
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)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x2, x3, x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(x5)
U23_AA(x1, x2, x3, x4, x5)  =  U23_AA(x5)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5, x6)  =  U2_AAAA(x6)
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA
U12_AAAA(x1, x2, x3, x4, x5, x6)  =  U12_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5, x6)  =  U26_AA(x6)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
P41_IN_AAAAA(x1, x2, x3, x4, x5)  =  P41_IN_AAAAA
U14_AAAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAAA(x6)
U15_AAAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAAA(x6)
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA
U13_AAAA(x1, x2, x3, x4, x5, x6)  =  U13_AAAA(x6)
U28_AA(x1, x2, x3)  =  U28_AA(x3)
QS84_IN_A(x1)  =  QS84_IN_A
U29_AA(x1, x2, x3)  =  U29_AA(x3)
U30_AA(x1, x2, x3)  =  U30_AA(x3)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x3, x4)
U32_AA(x1, x2, x3)  =  U32_AA(x3)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)
U13_GAGA(x1, x2, x3, x4, x5, x6)  =  U13_GAGA(x1, x2, x4, x6)
U21_AA(x1, x2, x3, x4, x5, x6)  =  U21_AA(x1, x6)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
P41_IN_AAAGA(x1, x2, x3, x4, x5)  =  P41_IN_AAAGA(x4)
U14_AAAGA(x1, x2, x3, x4, x5, x6)  =  U14_AAAGA(x4, x6)
U15_AAAGA(x1, x2, x3, x4, x5, x6)  =  U15_AAAGA(x4, x6)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)
U13_AGAA(x1, x2, x3, x4, x5, x6)  =  U13_AGAA(x3, x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x1, x2, x3, x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x1, x2, x3, x5)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5, x6)  =  U26_GA(x1, x2, x3, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3)  =  U28_GA(x1, x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x1, x3)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x1, x3, x4)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4, x5, x6)  =  U13_GGGA(x1, x2, x3, x4, x6)

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

(115) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U17_GA(T22, T23, T24, T9, less14_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → LESS14_IN_GG(T22, T23)
LESS14_IN_GG(s(T38), s(T39)) → U1_GG(T38, T39, less14_in_gg(T38, T39))
LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → PART24_IN_GGAA(T22, T24, X52, X53)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GG(T62, T63)
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_GGAA(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GGAA(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)
U17_GA(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → LESS14_IN_AA(T22, T23)
LESS14_IN_AA(s(T38), s(T39)) → U1_AA(T38, T39, less14_in_aa(T38, T39))
LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → PART24_IN_GAAA(T22, T24, X52, X53)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GA(T62, T63)
LESS14_IN_GA(s(T38), s(T39)) → U1_GA(T38, T39, less14_in_ga(T38, T39))
LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_GAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_GAAA(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)
U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U23_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
QS1_IN_AA(.(T191, .(T192, T193)), T9) → PART24_IN_AAAA(T191, T193, X336, X337)
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_AAAA(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
PART24_IN_AAAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_AA(T62, T63)
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_AAAA(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U2_AAAA(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → U4_AAAA(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)
QS1_IN_AA(.(T191, .(T192, T193)), T9) → U24_AA(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_AA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
QS42_IN_AA(.(T103, T104), X230) → U5_AA(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
QS42_IN_AA(.(T103, T104), X230) → PART24_IN_AAAA(T103, T104, X226, X227)
QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_AA(T103, T104, X230, qs42_in_aa(T108, X228))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_AA(T103, T104, X230, qs42_in_aa(T109, X229))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_AA(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_AA(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
U10_AA(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → APP57_IN_AAAA(T113, T103, T114, X230)
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → U12_AAAA(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)
U24_AA(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_AA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_AA(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
U26_AA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAAA(.(T192, T198), X15, T202, T191, T9)
P41_IN_AAAAA(T46, X15, T95, T22, T9) → U14_AAAAA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAAA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAAA(T46, T98, T95, T22, T9) → U15_AAAAA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAAA(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
U15_AAAAA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AAAA(T95, T22, T98, T9)
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AAAA(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)
QS1_IN_AA(.(T207, []), T9) → U28_AA(T207, T9, qs84_in_a(X14))
QS1_IN_AA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_AA(.(T207, []), T9) → U29_AA(T207, T9, qs84_in_a(T210))
U29_AA(T207, T9, qs84_out_a(T210)) → U30_AA(T207, T9, qs84_in_a(X15))
U29_AA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_AA(T207, T9, qs84_out_a(T210)) → U31_AA(T207, T9, T210, qs84_in_a(T213))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → U32_AA(T207, T9, app43_in_gaga(T210, T207, T213, T9))
U31_AA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GAGA(T210, T207, T213, T9)
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GAGA(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_AA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_AA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_AA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
P41_IN_AAAGA(T46, X15, T95, T22, T9) → U14_AAAGA(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
P41_IN_AAAGA(T46, X15, T95, T22, T9) → QS42_IN_AA(T46, X15)
P41_IN_AAAGA(T46, T98, T95, T22, T9) → U15_AAAGA(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_AAAGA(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
U15_AAAGA(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → APP43_IN_AGAA(T95, T22, T98, T9)
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → U13_AGAA(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)
U19_GA(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_GA(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_GA(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U21_GA(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → P41_IN_AAAGA(T46, X15, T95, T22, T9)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U23_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
QS1_IN_GA(.(T191, .(T192, T193)), T9) → PART24_IN_GGAA(T191, T193, X336, X337)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U24_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_GA(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → QS42_IN_AA(T197, X14)
U24_GA(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_GA(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_GA(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U26_GA(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → P41_IN_AAAGA(.(T192, T198), X15, T202, T191, T9)
QS1_IN_GA(.(T207, []), T9) → U28_GA(T207, T9, qs84_in_a(X14))
QS1_IN_GA(.(T207, []), T9) → QS84_IN_A(X14)
QS1_IN_GA(.(T207, []), T9) → U29_GA(T207, T9, qs84_in_a(T210))
U29_GA(T207, T9, qs84_out_a(T210)) → U30_GA(T207, T9, qs84_in_a(X15))
U29_GA(T207, T9, qs84_out_a(T210)) → QS84_IN_A(X15)
U29_GA(T207, T9, qs84_out_a(T210)) → U31_GA(T207, T9, T210, qs84_in_a(T213))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → U32_GA(T207, T9, app43_in_ggga(T210, T207, T213, T9))
U31_GA(T207, T9, T210, qs84_out_a(T213)) → APP43_IN_GGGA(T210, T207, T213, T9)
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → U13_GGGA(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)
LESS14_IN_GG(x1, x2)  =  LESS14_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x3, x5)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
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)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x2, x3, x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(x5)
U23_AA(x1, x2, x3, x4, x5)  =  U23_AA(x5)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5, x6)  =  U2_AAAA(x6)
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA
U12_AAAA(x1, x2, x3, x4, x5, x6)  =  U12_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5, x6)  =  U26_AA(x6)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
P41_IN_AAAAA(x1, x2, x3, x4, x5)  =  P41_IN_AAAAA
U14_AAAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAAA(x6)
U15_AAAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAAA(x6)
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA
U13_AAAA(x1, x2, x3, x4, x5, x6)  =  U13_AAAA(x6)
U28_AA(x1, x2, x3)  =  U28_AA(x3)
QS84_IN_A(x1)  =  QS84_IN_A
U29_AA(x1, x2, x3)  =  U29_AA(x3)
U30_AA(x1, x2, x3)  =  U30_AA(x3)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x3, x4)
U32_AA(x1, x2, x3)  =  U32_AA(x3)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)
U13_GAGA(x1, x2, x3, x4, x5, x6)  =  U13_GAGA(x1, x2, x4, x6)
U21_AA(x1, x2, x3, x4, x5, x6)  =  U21_AA(x1, x6)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
P41_IN_AAAGA(x1, x2, x3, x4, x5)  =  P41_IN_AAAGA(x4)
U14_AAAGA(x1, x2, x3, x4, x5, x6)  =  U14_AAAGA(x4, x6)
U15_AAAGA(x1, x2, x3, x4, x5, x6)  =  U15_AAAGA(x4, x6)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)
U13_AGAA(x1, x2, x3, x4, x5, x6)  =  U13_AGAA(x3, x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x1, x2, x3, x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x1, x2, x3, x5)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5, x6)  =  U26_GA(x1, x2, x3, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3)  =  U28_GA(x1, x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x1, x3)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x1, x3, x4)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4, x5, x6)  =  U13_GGGA(x1, x2, x3, x4, x6)

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

(116) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 13 SCCs with 85 less nodes.

(117) Complex Obligation (AND)

(118) Obligation:

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

APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_IN_GGGA(x1, x2, x3)

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

(119) UsableRulesProof (EQUIVALENT transformation)

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

(120) Obligation:

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

APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_GGGA(x1, x2, x3, x4)  =  APP43_IN_GGGA(x1, x2, x3)

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

(121) PiDPToQDPProof (SOUND transformation)

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

(122) Obligation:

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

APP43_IN_GGGA(.(T169, T170), T171, T172) → APP43_IN_GGGA(T170, T171, T172)

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

(123) 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:

• APP43_IN_GGGA(.(T169, T170), T171, T172) → APP43_IN_GGGA(T170, T171, T172)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(124) YES

(125) Obligation:

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

APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)

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

(126) UsableRulesProof (EQUIVALENT transformation)

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

(127) Obligation:

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

APP43_IN_AGAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AGAA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_AGAA(x1, x2, x3, x4)  =  APP43_IN_AGAA(x2)

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

(128) PiDPToQDPProof (SOUND transformation)

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

(129) Obligation:

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

APP43_IN_AGAA(T171) → APP43_IN_AGAA(T171)

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

(130) NonTerminationProof (EQUIVALENT transformation)

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

s = APP43_IN_AGAA(T171) evaluates to t =APP43_IN_AGAA(T171)

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP43_IN_AGAA(T171) to APP43_IN_AGAA(T171).

(131) NO

(132) Obligation:

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

APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)

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

(133) UsableRulesProof (EQUIVALENT transformation)

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

(134) Obligation:

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

APP43_IN_GAGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GAGA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_GAGA(x1, x2, x3, x4)  =  APP43_IN_GAGA(x1, x3)

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

(135) PiDPToQDPProof (SOUND transformation)

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

(136) Obligation:

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

APP43_IN_GAGA(.(T169, T170), T172) → APP43_IN_GAGA(T170, T172)

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

(137) 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:

• APP43_IN_GAGA(.(T169, T170), T172) → APP43_IN_GAGA(T170, T172)
  The graph contains the following edges 1 > 1, 2 >= 2

(138) YES

(139) Obligation:

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

APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA

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

(140) UsableRulesProof (EQUIVALENT transformation)

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

(141) Obligation:

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

APP43_IN_AAAA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_AAAA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP43_IN_AAAA(x1, x2, x3, x4)  =  APP43_IN_AAAA

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

(142) PiDPToQDPProof (SOUND transformation)

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

(143) Obligation:

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

APP43_IN_AAAA → APP43_IN_AAAA

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

(144) NonTerminationProof (EQUIVALENT transformation)

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

s = APP43_IN_AAAA evaluates to t =APP43_IN_AAAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP43_IN_AAAA to APP43_IN_AAAA.

(145) NO

(146) Obligation:

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

APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA

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

(147) UsableRulesProof (EQUIVALENT transformation)

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

(148) Obligation:

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

APP57_IN_AAAA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_AAAA(T138, T139, T140, X270)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP57_IN_AAAA(x1, x2, x3, x4)  =  APP57_IN_AAAA

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

(149) PiDPToQDPProof (SOUND transformation)

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

(150) Obligation:

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

APP57_IN_AAAA → APP57_IN_AAAA

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

(151) NonTerminationProof (EQUIVALENT transformation)

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

s = APP57_IN_AAAA evaluates to t =APP57_IN_AAAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP57_IN_AAAA to APP57_IN_AAAA.

(152) NO

(153) Obligation:

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

LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)

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

(154) UsableRulesProof (EQUIVALENT transformation)

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

(155) Obligation:

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

LESS14_IN_GA(s(T38), s(T39)) → LESS14_IN_GA(T38, T39)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS14_IN_GA(x1, x2)  =  LESS14_IN_GA(x1)

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

(156) PiDPToQDPProof (SOUND transformation)

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

(157) Obligation:

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

LESS14_IN_GA(s(T38)) → LESS14_IN_GA(T38)

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

(158) 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:

• LESS14_IN_GA(s(T38)) → LESS14_IN_GA(T38)
  The graph contains the following edges 1 > 1

(159) YES

(160) Obligation:

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

U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)

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

(161) UsableRulesProof (EQUIVALENT transformation)

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

(162) Obligation:

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

U2_GAAA(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → PART24_IN_GAAA(T62, T64, X126, X127)
PART24_IN_GAAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GAAA(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
PART24_IN_GAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GAAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
PART24_IN_GAAA(x1, x2, x3, x4)  =  PART24_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)

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

(163) PiDPToQDPProof (SOUND transformation)

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

(164) Obligation:

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

U2_GAAA(T62, less14_out_ga(T62)) → PART24_IN_GAAA(T62)
PART24_IN_GAAA(T62) → U2_GAAA(T62, less14_in_ga(T62))
PART24_IN_GAAA(T84) → PART24_IN_GAAA(T84)

The TRS R consists of the following rules:

less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))

The set Q consists of the following terms:

less14_in_ga(x0)
U1_ga(x0, x1)

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

(165) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART24_IN_GAAA(T62) → U2_GAAA(T62, less14_in_ga(T62)) at position [1] we obtained the following new rules [LPAR04]:

PART24_IN_GAAA(0) → U2_GAAA(0, less14_out_ga(0))
PART24_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, less14_in_ga(x0)))

(166) Obligation:

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

U2_GAAA(T62, less14_out_ga(T62)) → PART24_IN_GAAA(T62)
PART24_IN_GAAA(T84) → PART24_IN_GAAA(T84)
PART24_IN_GAAA(0) → U2_GAAA(0, less14_out_ga(0))
PART24_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, less14_in_ga(x0)))

The TRS R consists of the following rules:

less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))

The set Q consists of the following terms:

less14_in_ga(x0)
U1_ga(x0, x1)

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

(167) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GAAA(T62, less14_out_ga(T62)) → PART24_IN_GAAA(T62) we obtained the following new rules [LPAR04]:

U2_GAAA(0, less14_out_ga(0)) → PART24_IN_GAAA(0)
U2_GAAA(s(z0), less14_out_ga(s(z0))) → PART24_IN_GAAA(s(z0))

(168) Obligation:

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

PART24_IN_GAAA(T84) → PART24_IN_GAAA(T84)
PART24_IN_GAAA(0) → U2_GAAA(0, less14_out_ga(0))
PART24_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, less14_in_ga(x0)))
U2_GAAA(0, less14_out_ga(0)) → PART24_IN_GAAA(0)
U2_GAAA(s(z0), less14_out_ga(s(z0))) → PART24_IN_GAAA(s(z0))

The TRS R consists of the following rules:

less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))

The set Q consists of the following terms:

less14_in_ga(x0)
U1_ga(x0, x1)

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

(169) NonTerminationProof (EQUIVALENT transformation)

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

s = PART24_IN_GAAA(T84) evaluates to t =PART24_IN_GAAA(T84)

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART24_IN_GAAA(T84) to PART24_IN_GAAA(T84).

(170) NO

(171) Obligation:

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

LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA

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

(172) UsableRulesProof (EQUIVALENT transformation)

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

(173) Obligation:

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

LESS14_IN_AA(s(T38), s(T39)) → LESS14_IN_AA(T38, T39)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS14_IN_AA(x1, x2)  =  LESS14_IN_AA

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

(174) PiDPToQDPProof (SOUND transformation)

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

(175) Obligation:

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

LESS14_IN_AA → LESS14_IN_AA

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

(176) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS14_IN_AA evaluates to t =LESS14_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS14_IN_AA to LESS14_IN_AA.

(177) NO

(178) Obligation:

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

PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA

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

(179) UsableRulesProof (EQUIVALENT transformation)

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

(180) Obligation:

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

PART24_IN_AAAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_AAAA(T84, T86, X177, X178)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
PART24_IN_AAAA(x1, x2, x3, x4)  =  PART24_IN_AAAA

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

(181) PiDPToQDPProof (SOUND transformation)

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

(182) Obligation:

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

PART24_IN_AAAA → PART24_IN_AAAA

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

(183) NonTerminationProof (EQUIVALENT transformation)

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

s = PART24_IN_AAAA evaluates to t =PART24_IN_AAAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART24_IN_AAAA to PART24_IN_AAAA.

(184) NO

(185) Obligation:

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

QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)

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

(186) UsableRulesProof (EQUIVALENT transformation)

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

(187) Obligation:

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

QS42_IN_AA(.(T103, T104), X230) → U6_AA(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → QS42_IN_AA(T108, X228)
U6_AA(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_AA(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_AA(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → QS42_IN_AA(T109, X229)

The TRS R consists of the following rules:

part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
QS42_IN_AA(x1, x2)  =  QS42_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)

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

(188) PiDPToQDPProof (SOUND transformation)

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

(189) Obligation:

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

QS42_IN_AA → U6_AA(part24_in_aaaa)
U6_AA(part24_out_aaaa) → QS42_IN_AA
U6_AA(part24_out_aaaa) → U8_AA(qs42_in_aa)
U8_AA(qs42_out_aa) → QS42_IN_AA

The TRS R consists of the following rules:

part24_in_aaaa → U2_aaaa(less14_in_aa)
part24_in_aaaa → U4_aaaa(part24_in_aaaa)
part24_in_aaaa → part24_out_aaaa
qs42_in_aa → qs42_out_aa
qs42_in_aa → U5_aa(part24_in_aaaa)
qs42_in_aa → U6_aa(part24_in_aaaa)
U2_aaaa(less14_out_aa(T62)) → part24_out_aaaa
U2_aaaa(less14_out_aa(T62)) → U3_aaaa(part24_in_gaaa(T62))
U4_aaaa(part24_out_aaaa) → part24_out_aaaa
U5_aa(part24_out_aaaa) → qs42_out_aa
U6_aa(part24_out_aaaa) → U7_aa(qs42_in_aa)
U6_aa(part24_out_aaaa) → U8_aa(qs42_in_aa)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U3_aaaa(part24_out_gaaa(T62)) → part24_out_aaaa
U7_aa(qs42_out_aa) → qs42_out_aa
U8_aa(qs42_out_aa) → U9_aa(qs42_in_aa)
U8_aa(qs42_out_aa) → U10_aa(qs42_in_aa)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(T84, part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa(T92)
U9_aa(qs42_out_aa) → qs42_out_aa
U10_aa(qs42_out_aa) → U11_aa(app57_in_aaaa)
U2_gaaa(T62, less14_out_ga(T62)) → part24_out_gaaa(T62)
U2_gaaa(T62, less14_out_ga(T62)) → U3_gaaa(T62, part24_in_gaaa(T62))
U4_gaaa(T84, part24_out_gaaa(T84)) → part24_out_gaaa(T84)
U11_aa(app57_out_aaaa) → qs42_out_aa
less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U3_gaaa(T62, part24_out_gaaa(T62)) → part24_out_gaaa(T62)
app57_in_aaaa → app57_out_aaaa
app57_in_aaaa → U12_aaaa(app57_in_aaaa)
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))
U12_aaaa(app57_out_aaaa) → app57_out_aaaa

The set Q consists of the following terms:

part24_in_aaaa
qs42_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U5_aa(x0)
U6_aa(x0)
less14_in_aa
U3_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
U1_aa(x0)
part24_in_gaaa(x0)
U9_aa(x0)
U10_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U11_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0, x1)
app57_in_aaaa
U1_ga(x0, x1)
U12_aaaa(x0)

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

(190) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS42_IN_AA → U6_AA(part24_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS42_IN_AA → U6_AA(U2_aaaa(less14_in_aa))
QS42_IN_AA → U6_AA(U4_aaaa(part24_in_aaaa))
QS42_IN_AA → U6_AA(part24_out_aaaa)

(191) Obligation:

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

U6_AA(part24_out_aaaa) → QS42_IN_AA
U6_AA(part24_out_aaaa) → U8_AA(qs42_in_aa)
U8_AA(qs42_out_aa) → QS42_IN_AA
QS42_IN_AA → U6_AA(U2_aaaa(less14_in_aa))
QS42_IN_AA → U6_AA(U4_aaaa(part24_in_aaaa))
QS42_IN_AA → U6_AA(part24_out_aaaa)

The TRS R consists of the following rules:

part24_in_aaaa → U2_aaaa(less14_in_aa)
part24_in_aaaa → U4_aaaa(part24_in_aaaa)
part24_in_aaaa → part24_out_aaaa
qs42_in_aa → qs42_out_aa
qs42_in_aa → U5_aa(part24_in_aaaa)
qs42_in_aa → U6_aa(part24_in_aaaa)
U2_aaaa(less14_out_aa(T62)) → part24_out_aaaa
U2_aaaa(less14_out_aa(T62)) → U3_aaaa(part24_in_gaaa(T62))
U4_aaaa(part24_out_aaaa) → part24_out_aaaa
U5_aa(part24_out_aaaa) → qs42_out_aa
U6_aa(part24_out_aaaa) → U7_aa(qs42_in_aa)
U6_aa(part24_out_aaaa) → U8_aa(qs42_in_aa)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U3_aaaa(part24_out_gaaa(T62)) → part24_out_aaaa
U7_aa(qs42_out_aa) → qs42_out_aa
U8_aa(qs42_out_aa) → U9_aa(qs42_in_aa)
U8_aa(qs42_out_aa) → U10_aa(qs42_in_aa)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(T84, part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa(T92)
U9_aa(qs42_out_aa) → qs42_out_aa
U10_aa(qs42_out_aa) → U11_aa(app57_in_aaaa)
U2_gaaa(T62, less14_out_ga(T62)) → part24_out_gaaa(T62)
U2_gaaa(T62, less14_out_ga(T62)) → U3_gaaa(T62, part24_in_gaaa(T62))
U4_gaaa(T84, part24_out_gaaa(T84)) → part24_out_gaaa(T84)
U11_aa(app57_out_aaaa) → qs42_out_aa
less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U3_gaaa(T62, part24_out_gaaa(T62)) → part24_out_gaaa(T62)
app57_in_aaaa → app57_out_aaaa
app57_in_aaaa → U12_aaaa(app57_in_aaaa)
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))
U12_aaaa(app57_out_aaaa) → app57_out_aaaa

The set Q consists of the following terms:

part24_in_aaaa
qs42_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U5_aa(x0)
U6_aa(x0)
less14_in_aa
U3_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
U1_aa(x0)
part24_in_gaaa(x0)
U9_aa(x0)
U10_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U11_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0, x1)
app57_in_aaaa
U1_ga(x0, x1)
U12_aaaa(x0)

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

(192) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U6_AA(part24_out_aaaa) → U8_AA(qs42_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U6_AA(part24_out_aaaa) → U8_AA(qs42_out_aa)
U6_AA(part24_out_aaaa) → U8_AA(U5_aa(part24_in_aaaa))
U6_AA(part24_out_aaaa) → U8_AA(U6_aa(part24_in_aaaa))

(193) Obligation:

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

U6_AA(part24_out_aaaa) → QS42_IN_AA
U8_AA(qs42_out_aa) → QS42_IN_AA
QS42_IN_AA → U6_AA(U2_aaaa(less14_in_aa))
QS42_IN_AA → U6_AA(U4_aaaa(part24_in_aaaa))
QS42_IN_AA → U6_AA(part24_out_aaaa)
U6_AA(part24_out_aaaa) → U8_AA(qs42_out_aa)
U6_AA(part24_out_aaaa) → U8_AA(U5_aa(part24_in_aaaa))
U6_AA(part24_out_aaaa) → U8_AA(U6_aa(part24_in_aaaa))

The TRS R consists of the following rules:

part24_in_aaaa → U2_aaaa(less14_in_aa)
part24_in_aaaa → U4_aaaa(part24_in_aaaa)
part24_in_aaaa → part24_out_aaaa
qs42_in_aa → qs42_out_aa
qs42_in_aa → U5_aa(part24_in_aaaa)
qs42_in_aa → U6_aa(part24_in_aaaa)
U2_aaaa(less14_out_aa(T62)) → part24_out_aaaa
U2_aaaa(less14_out_aa(T62)) → U3_aaaa(part24_in_gaaa(T62))
U4_aaaa(part24_out_aaaa) → part24_out_aaaa
U5_aa(part24_out_aaaa) → qs42_out_aa
U6_aa(part24_out_aaaa) → U7_aa(qs42_in_aa)
U6_aa(part24_out_aaaa) → U8_aa(qs42_in_aa)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U3_aaaa(part24_out_gaaa(T62)) → part24_out_aaaa
U7_aa(qs42_out_aa) → qs42_out_aa
U8_aa(qs42_out_aa) → U9_aa(qs42_in_aa)
U8_aa(qs42_out_aa) → U10_aa(qs42_in_aa)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(T84, part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa(T92)
U9_aa(qs42_out_aa) → qs42_out_aa
U10_aa(qs42_out_aa) → U11_aa(app57_in_aaaa)
U2_gaaa(T62, less14_out_ga(T62)) → part24_out_gaaa(T62)
U2_gaaa(T62, less14_out_ga(T62)) → U3_gaaa(T62, part24_in_gaaa(T62))
U4_gaaa(T84, part24_out_gaaa(T84)) → part24_out_gaaa(T84)
U11_aa(app57_out_aaaa) → qs42_out_aa
less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U3_gaaa(T62, part24_out_gaaa(T62)) → part24_out_gaaa(T62)
app57_in_aaaa → app57_out_aaaa
app57_in_aaaa → U12_aaaa(app57_in_aaaa)
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))
U12_aaaa(app57_out_aaaa) → app57_out_aaaa

The set Q consists of the following terms:

part24_in_aaaa
qs42_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U5_aa(x0)
U6_aa(x0)
less14_in_aa
U3_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
U1_aa(x0)
part24_in_gaaa(x0)
U9_aa(x0)
U10_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U11_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0, x1)
app57_in_aaaa
U1_ga(x0, x1)
U12_aaaa(x0)

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

(194) NonTerminationProof (EQUIVALENT transformation)

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

s = QS42_IN_AA evaluates to t =QS42_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS42_IN_AA → U6_AA(part24_out_aaaa)
with rule QS42_IN_AA → U6_AA(part24_out_aaaa) at position [] and matcher [ ]

U6_AA(part24_out_aaaa) → QS42_IN_AA
with rule U6_AA(part24_out_aaaa) → QS42_IN_AA

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


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

(195) NO

(196) Obligation:

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

U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)

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

(197) UsableRulesProof (EQUIVALENT transformation)

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

(198) Obligation:

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

U17_AA(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_AA(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_AA(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → QS1_IN_AA(.(T23, T45), X14)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U17_AA(T22, T23, T24, T9, less14_in_aa(T22, T23))

The TRS R consists of the following rules:

part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x1, x5)

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

(199) PiDPToQDPProof (SOUND transformation)

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

(200) Obligation:

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

U17_AA(less14_out_aa(T22)) → U19_AA(T22, part24_in_gaaa(T22))
U19_AA(T22, part24_out_gaaa(T22)) → QS1_IN_AA
QS1_IN_AA → U17_AA(less14_in_aa)

The TRS R consists of the following rules:

part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(T84, part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa(T92)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U2_gaaa(T62, less14_out_ga(T62)) → part24_out_gaaa(T62)
U2_gaaa(T62, less14_out_ga(T62)) → U3_gaaa(T62, part24_in_gaaa(T62))
U4_gaaa(T84, part24_out_gaaa(T84)) → part24_out_gaaa(T84)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U3_gaaa(T62, part24_out_gaaa(T62)) → part24_out_gaaa(T62)
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))

The set Q consists of the following terms:

part24_in_gaaa(x0)
less14_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0, x1)
U1_ga(x0, x1)

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

(201) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U17_AA(less14_out_aa(T22)) → U19_AA(T22, part24_in_gaaa(T22)) at position [1] we obtained the following new rules [LPAR04]:

U17_AA(less14_out_aa(x0)) → U19_AA(x0, U2_gaaa(x0, less14_in_ga(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U4_gaaa(x0, part24_in_gaaa(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa(x0))

(202) Obligation:

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

U19_AA(T22, part24_out_gaaa(T22)) → QS1_IN_AA
QS1_IN_AA → U17_AA(less14_in_aa)
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U2_gaaa(x0, less14_in_ga(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U4_gaaa(x0, part24_in_gaaa(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa(x0))

The TRS R consists of the following rules:

part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(T84, part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa(T92)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U2_gaaa(T62, less14_out_ga(T62)) → part24_out_gaaa(T62)
U2_gaaa(T62, less14_out_ga(T62)) → U3_gaaa(T62, part24_in_gaaa(T62))
U4_gaaa(T84, part24_out_gaaa(T84)) → part24_out_gaaa(T84)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U3_gaaa(T62, part24_out_gaaa(T62)) → part24_out_gaaa(T62)
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))

The set Q consists of the following terms:

part24_in_gaaa(x0)
less14_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0, x1)
U1_ga(x0, x1)

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

(203) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS1_IN_AA → U17_AA(less14_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QS1_IN_AA → U17_AA(less14_out_aa(0))
QS1_IN_AA → U17_AA(U1_aa(less14_in_aa))

(204) Obligation:

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

U19_AA(T22, part24_out_gaaa(T22)) → QS1_IN_AA
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U2_gaaa(x0, less14_in_ga(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, U4_gaaa(x0, part24_in_gaaa(x0)))
U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa(x0))
QS1_IN_AA → U17_AA(less14_out_aa(0))
QS1_IN_AA → U17_AA(U1_aa(less14_in_aa))

The TRS R consists of the following rules:

part24_in_gaaa(T62) → U2_gaaa(T62, less14_in_ga(T62))
part24_in_gaaa(T84) → U4_gaaa(T84, part24_in_gaaa(T84))
part24_in_gaaa(T92) → part24_out_gaaa(T92)
less14_in_aa → less14_out_aa(0)
less14_in_aa → U1_aa(less14_in_aa)
U2_gaaa(T62, less14_out_ga(T62)) → part24_out_gaaa(T62)
U2_gaaa(T62, less14_out_ga(T62)) → U3_gaaa(T62, part24_in_gaaa(T62))
U4_gaaa(T84, part24_out_gaaa(T84)) → part24_out_gaaa(T84)
U1_aa(less14_out_aa(T38)) → less14_out_aa(s(T38))
less14_in_ga(0) → less14_out_ga(0)
less14_in_ga(s(T38)) → U1_ga(T38, less14_in_ga(T38))
U3_gaaa(T62, part24_out_gaaa(T62)) → part24_out_gaaa(T62)
U1_ga(T38, less14_out_ga(T38)) → less14_out_ga(s(T38))

The set Q consists of the following terms:

part24_in_gaaa(x0)
less14_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
less14_in_ga(x0)
U3_gaaa(x0, x1)
U1_ga(x0, x1)

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

(205) NonTerminationProof (EQUIVALENT transformation)

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

s = QS1_IN_AA evaluates to t =QS1_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS1_IN_AA → U17_AA(less14_out_aa(0))
with rule QS1_IN_AA → U17_AA(less14_out_aa(0)) at position [] and matcher [ ]

U17_AA(less14_out_aa(0)) → U19_AA(0, part24_out_gaaa(0))
with rule U17_AA(less14_out_aa(x0)) → U19_AA(x0, part24_out_gaaa(x0)) at position [] and matcher [x0 / 0]

U19_AA(0, part24_out_gaaa(0)) → QS1_IN_AA
with rule U19_AA(T22, part24_out_gaaa(T22)) → QS1_IN_AA

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


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

(206) NO

(207) Obligation:

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

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
LESS14_IN_GG(x1, x2)  =  LESS14_IN_GG(x1, x2)

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

(208) UsableRulesProof (EQUIVALENT transformation)

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

(209) Obligation:

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

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

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

(210) PiDPToQDPProof (EQUIVALENT transformation)

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

(211) Obligation:

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

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

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

(212) 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:

• LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
  The graph contains the following edges 1 > 1, 2 > 2

(213) YES

(214) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(.(T22, .(T23, T24)), T9) → U17_ga(T22, T23, T24, T9, less14_in_gg(T22, T23))
less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U18_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
part24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U2_ggaa(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U2_ggaa(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → U3_ggaa(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
part24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_ggaa(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
part24_in_ggaa(T92, [], [], []) → part24_out_ggaa(T92, [], [], [])
U4_ggaa(T84, T85, T86, X177, X178, part24_out_ggaa(T84, T86, X177, X178)) → part24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U3_ggaa(T62, T63, T64, X126, X127, part24_out_ggaa(T62, T64, X126, X127)) → part24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U18_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, X52, X53)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U17_ga(T22, T23, T24, T9, less14_out_gg(T22, T23)) → U19_ga(T22, T23, T24, T9, part24_in_ggaa(T22, T24, T45, T46))
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U20_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(.(T22, .(T23, T24)), T9) → U17_aa(T22, T23, T24, T9, less14_in_aa(T22, T23))
less14_in_aa(0, s(T33)) → less14_out_aa(0, s(T33))
less14_in_aa(s(T38), s(T39)) → U1_aa(T38, T39, less14_in_aa(T38, T39))
U1_aa(T38, T39, less14_out_aa(T38, T39)) → less14_out_aa(s(T38), s(T39))
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U18_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, X52, X53))
part24_in_gaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_gaaa(T62, T63, T64, X126, X127, less14_in_ga(T62, T63))
less14_in_ga(0, s(T33)) → less14_out_ga(0, s(T33))
less14_in_ga(s(T38), s(T39)) → U1_ga(T38, T39, less14_in_ga(T38, T39))
U1_ga(T38, T39, less14_out_ga(T38, T39)) → less14_out_ga(s(T38), s(T39))
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_gaaa(T62, T63, T64, X126, X127, less14_out_ga(T62, T63)) → U3_gaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
part24_in_gaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_gaaa(T84, T85, T86, X177, X178, part24_in_gaaa(T84, T86, X177, X178))
part24_in_gaaa(T92, [], [], []) → part24_out_gaaa(T92, [], [], [])
U4_gaaa(T84, T85, T86, X177, X178, part24_out_gaaa(T84, T86, X177, X178)) → part24_out_gaaa(T84, .(T85, T86), X177, .(T85, X178))
U3_gaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_gaaa(T62, .(T63, T64), .(T63, X126), X127)
U18_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, X52, X53)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U17_aa(T22, T23, T24, T9, less14_out_aa(T22, T23)) → U19_aa(T22, T23, T24, T9, part24_in_gaaa(T22, T24, T45, T46))
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U20_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T45), X14))
qs1_in_aa(.(T191, .(T192, T193)), T9) → U23_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, X336, X337))
part24_in_aaaa(T62, .(T63, T64), .(T63, X126), X127) → U2_aaaa(T62, T63, T64, X126, X127, less14_in_aa(T62, T63))
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
U2_aaaa(T62, T63, T64, X126, X127, less14_out_aa(T62, T63)) → U3_aaaa(T62, T63, T64, X126, X127, part24_in_gaaa(T62, T64, X126, X127))
U3_aaaa(T62, T63, T64, X126, X127, part24_out_gaaa(T62, T64, X126, X127)) → part24_out_aaaa(T62, .(T63, T64), .(T63, X126), X127)
part24_in_aaaa(T84, .(T85, T86), X177, .(T85, X178)) → U4_aaaa(T84, T85, T86, X177, X178, part24_in_aaaa(T84, T86, X177, X178))
part24_in_aaaa(T92, [], [], []) → part24_out_aaaa(T92, [], [], [])
U4_aaaa(T84, T85, T86, X177, X178, part24_out_aaaa(T84, T86, X177, X178)) → part24_out_aaaa(T84, .(T85, T86), X177, .(T85, X178))
U23_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, X336, X337)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T191, .(T192, T193)), T9) → U24_aa(T191, T192, T193, T9, part24_in_aaaa(T191, T193, T197, T198))
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U25_aa(T191, T192, T193, T9, qs42_in_aa(T197, X14))
qs42_in_aa([], []) → qs42_out_aa([], [])
qs42_in_aa(.(T103, T104), X230) → U5_aa(T103, T104, X230, part24_in_aaaa(T103, T104, X226, X227))
U5_aa(T103, T104, X230, part24_out_aaaa(T103, T104, X226, X227)) → qs42_out_aa(.(T103, T104), X230)
qs42_in_aa(.(T103, T104), X230) → U6_aa(T103, T104, X230, part24_in_aaaa(T103, T104, T108, T109))
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U7_aa(T103, T104, X230, qs42_in_aa(T108, X228))
U7_aa(T103, T104, X230, qs42_out_aa(T108, X228)) → qs42_out_aa(.(T103, T104), X230)
U6_aa(T103, T104, X230, part24_out_aaaa(T103, T104, T108, T109)) → U8_aa(T103, T104, X230, T109, qs42_in_aa(T108, T113))
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U9_aa(T103, T104, X230, qs42_in_aa(T109, X229))
U9_aa(T103, T104, X230, qs42_out_aa(T109, X229)) → qs42_out_aa(.(T103, T104), X230)
U8_aa(T103, T104, X230, T109, qs42_out_aa(T108, T113)) → U10_aa(T103, T104, X230, T113, qs42_in_aa(T109, T114))
U10_aa(T103, T104, X230, T113, qs42_out_aa(T109, T114)) → U11_aa(T103, T104, X230, app57_in_aaaa(T113, T103, T114, X230))
app57_in_aaaa([], T127, T128, .(T127, T128)) → app57_out_aaaa([], T127, T128, .(T127, T128))
app57_in_aaaa(.(T137, T138), T139, T140, .(T137, X270)) → U12_aaaa(T137, T138, T139, T140, X270, app57_in_aaaa(T138, T139, T140, X270))
U12_aaaa(T137, T138, T139, T140, X270, app57_out_aaaa(T138, T139, T140, X270)) → app57_out_aaaa(.(T137, T138), T139, T140, .(T137, X270))
U11_aa(T103, T104, X230, app57_out_aaaa(T113, T103, T114, X230)) → qs42_out_aa(.(T103, T104), X230)
U25_aa(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
U24_aa(T191, T192, T193, T9, part24_out_aaaa(T191, T193, T197, T198)) → U26_aa(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_aa(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_aa(T191, T192, T193, T9, p41_in_aaaaa(.(T192, T198), X15, T202, T191, T9))
p41_in_aaaaa(T46, X15, T95, T22, T9) → U14_aaaaa(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaaa(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaaa(T46, X15, T95, T22, T9)
p41_in_aaaaa(T46, T98, T95, T22, T9) → U15_aaaaa(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaaa(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaaa(T46, T98, T95, T22, T9, app43_in_aaaa(T95, T22, T98, T9))
app43_in_aaaa([], T157, T158, .(T157, T158)) → app43_out_aaaa([], T157, T158, .(T157, T158))
app43_in_aaaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_aaaa(T169, T170, T171, T172, T174, app43_in_aaaa(T170, T171, T172, T174))
U13_aaaa(T169, T170, T171, T172, T174, app43_out_aaaa(T170, T171, T172, T174)) → app43_out_aaaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaaa(T46, T98, T95, T22, T9, app43_out_aaaa(T95, T22, T98, T9)) → p41_out_aaaaa(T46, T98, T95, T22, T9)
U27_aa(T191, T192, T193, T9, p41_out_aaaaa(.(T192, T198), X15, T202, T191, T9)) → qs1_out_aa(.(T191, .(T192, T193)), T9)
qs1_in_aa(.(T207, []), T9) → U28_aa(T207, T9, qs84_in_a(X14))
qs84_in_a([]) → qs84_out_a([])
U28_aa(T207, T9, qs84_out_a(X14)) → qs1_out_aa(.(T207, []), T9)
qs1_in_aa(.(T207, []), T9) → U29_aa(T207, T9, qs84_in_a(T210))
U29_aa(T207, T9, qs84_out_a(T210)) → U30_aa(T207, T9, qs84_in_a(X15))
U30_aa(T207, T9, qs84_out_a(X15)) → qs1_out_aa(.(T207, []), T9)
U29_aa(T207, T9, qs84_out_a(T210)) → U31_aa(T207, T9, T210, qs84_in_a(T213))
U31_aa(T207, T9, T210, qs84_out_a(T213)) → U32_aa(T207, T9, app43_in_gaga(T210, T207, T213, T9))
app43_in_gaga([], T157, T158, .(T157, T158)) → app43_out_gaga([], T157, T158, .(T157, T158))
app43_in_gaga(.(T169, T170), T171, T172, .(T169, T174)) → U13_gaga(T169, T170, T171, T172, T174, app43_in_gaga(T170, T171, T172, T174))
U13_gaga(T169, T170, T171, T172, T174, app43_out_gaga(T170, T171, T172, T174)) → app43_out_gaga(.(T169, T170), T171, T172, .(T169, T174))
U32_aa(T207, T9, app43_out_gaga(T210, T207, T213, T9)) → qs1_out_aa(.(T207, []), T9)
U20_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, part24_out_gaaa(T22, T24, T45, T46)) → U21_aa(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_aa(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_aa(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
p41_in_aaaga(T46, X15, T95, T22, T9) → U14_aaaga(T46, X15, T95, T22, T9, qs42_in_aa(T46, X15))
U14_aaaga(T46, X15, T95, T22, T9, qs42_out_aa(T46, X15)) → p41_out_aaaga(T46, X15, T95, T22, T9)
p41_in_aaaga(T46, T98, T95, T22, T9) → U15_aaaga(T46, T98, T95, T22, T9, qs42_in_aa(T46, T98))
U15_aaaga(T46, T98, T95, T22, T9, qs42_out_aa(T46, T98)) → U16_aaaga(T46, T98, T95, T22, T9, app43_in_agaa(T95, T22, T98, T9))
app43_in_agaa([], T157, T158, .(T157, T158)) → app43_out_agaa([], T157, T158, .(T157, T158))
app43_in_agaa(.(T169, T170), T171, T172, .(T169, T174)) → U13_agaa(T169, T170, T171, T172, T174, app43_in_agaa(T170, T171, T172, T174))
U13_agaa(T169, T170, T171, T172, T174, app43_out_agaa(T170, T171, T172, T174)) → app43_out_agaa(.(T169, T170), T171, T172, .(T169, T174))
U16_aaaga(T46, T98, T95, T22, T9, app43_out_agaa(T95, T22, T98, T9)) → p41_out_aaaga(T46, T98, T95, T22, T9)
U22_aa(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U20_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T45), X14)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, part24_out_ggaa(T22, T24, T45, T46)) → U21_ga(T22, T23, T24, T9, T46, qs1_in_aa(.(T23, T45), T95))
U21_ga(T22, T23, T24, T9, T46, qs1_out_aa(.(T23, T45), T95)) → U22_ga(T22, T23, T24, T9, p41_in_aaaga(T46, X15, T95, T22, T9))
U22_ga(T22, T23, T24, T9, p41_out_aaaga(T46, X15, T95, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U23_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
U23_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, X336, X337)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T191, .(T192, T193)), T9) → U24_ga(T191, T192, T193, T9, part24_in_ggaa(T191, T193, T197, T198))
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U25_ga(T191, T192, T193, T9, qs42_in_aa(T197, X14))
U25_ga(T191, T192, T193, T9, qs42_out_aa(T197, X14)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
U24_ga(T191, T192, T193, T9, part24_out_ggaa(T191, T193, T197, T198)) → U26_ga(T191, T192, T193, T9, T198, qs42_in_aa(T197, T202))
U26_ga(T191, T192, T193, T9, T198, qs42_out_aa(T197, T202)) → U27_ga(T191, T192, T193, T9, p41_in_aaaga(.(T192, T198), X15, T202, T191, T9))
U27_ga(T191, T192, T193, T9, p41_out_aaaga(.(T192, T198), X15, T202, T191, T9)) → qs1_out_ga(.(T191, .(T192, T193)), T9)
qs1_in_ga(.(T207, []), T9) → U28_ga(T207, T9, qs84_in_a(X14))
U28_ga(T207, T9, qs84_out_a(X14)) → qs1_out_ga(.(T207, []), T9)
qs1_in_ga(.(T207, []), T9) → U29_ga(T207, T9, qs84_in_a(T210))
U29_ga(T207, T9, qs84_out_a(T210)) → U30_ga(T207, T9, qs84_in_a(X15))
U30_ga(T207, T9, qs84_out_a(X15)) → qs1_out_ga(.(T207, []), T9)
U29_ga(T207, T9, qs84_out_a(T210)) → U31_ga(T207, T9, T210, qs84_in_a(T213))
U31_ga(T207, T9, T210, qs84_out_a(T213)) → U32_ga(T207, T9, app43_in_ggga(T210, T207, T213, T9))
app43_in_ggga([], T157, T158, .(T157, T158)) → app43_out_ggga([], T157, T158, .(T157, T158))
app43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U13_ggga(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
U13_ggga(T169, T170, T171, T172, T174, app43_out_ggga(T170, T171, T172, T174)) → app43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U32_ga(T207, T9, app43_out_ggga(T210, T207, T213, T9)) → qs1_out_ga(.(T207, []), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
part24_in_ggaa(x1, x2, x3, x4)  =  part24_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part24_out_ggaa(x1, x2, x3, x4)  =  part24_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
less14_in_aa(x1, x2)  =  less14_in_aa
less14_out_aa(x1, x2)  =  less14_out_aa(x1)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
part24_in_gaaa(x1, x2, x3, x4)  =  part24_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
less14_in_ga(x1, x2)  =  less14_in_ga(x1)
less14_out_ga(x1, x2)  =  less14_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
part24_out_gaaa(x1, x2, x3, x4)  =  part24_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(x1, x5)
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(x5)
U23_aa(x1, x2, x3, x4, x5)  =  U23_aa(x5)
part24_in_aaaa(x1, x2, x3, x4)  =  part24_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part24_out_aaaa(x1, x2, x3, x4)  =  part24_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
qs42_in_aa(x1, x2)  =  qs42_in_aa
qs42_out_aa(x1, x2)  =  qs42_out_aa
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
app57_in_aaaa(x1, x2, x3, x4)  =  app57_in_aaaa
app57_out_aaaa(x1, x2, x3, x4)  =  app57_out_aaaa
U12_aaaa(x1, x2, x3, x4, x5, x6)  =  U12_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5, x6)  =  U26_aa(x6)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
p41_in_aaaaa(x1, x2, x3, x4, x5)  =  p41_in_aaaaa
U14_aaaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaaa(x6)
p41_out_aaaaa(x1, x2, x3, x4, x5)  =  p41_out_aaaaa
U15_aaaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaaa(x6)
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
app43_in_aaaa(x1, x2, x3, x4)  =  app43_in_aaaa
app43_out_aaaa(x1, x2, x3, x4)  =  app43_out_aaaa
U13_aaaa(x1, x2, x3, x4, x5, x6)  =  U13_aaaa(x6)
U28_aa(x1, x2, x3)  =  U28_aa(x3)
qs84_in_a(x1)  =  qs84_in_a
qs84_out_a(x1)  =  qs84_out_a(x1)
U29_aa(x1, x2, x3)  =  U29_aa(x3)
U30_aa(x1, x2, x3)  =  U30_aa(x3)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x3, x4)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
app43_in_gaga(x1, x2, x3, x4)  =  app43_in_gaga(x1, x3)
app43_out_gaga(x1, x2, x3, x4)  =  app43_out_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5, x6)  =  U13_gaga(x1, x2, x4, x6)
U21_aa(x1, x2, x3, x4, x5, x6)  =  U21_aa(x1, x6)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
p41_in_aaaga(x1, x2, x3, x4, x5)  =  p41_in_aaaga(x4)
U14_aaaga(x1, x2, x3, x4, x5, x6)  =  U14_aaaga(x4, x6)
p41_out_aaaga(x1, x2, x3, x4, x5)  =  p41_out_aaaga(x4)
U15_aaaga(x1, x2, x3, x4, x5, x6)  =  U15_aaaga(x4, x6)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
app43_in_agaa(x1, x2, x3, x4)  =  app43_in_agaa(x2)
app43_out_agaa(x1, x2, x3, x4)  =  app43_out_agaa(x2)
U13_agaa(x1, x2, x3, x4, x5, x6)  =  U13_agaa(x3, x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x1, x2, x3, x5)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x6)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x3, x4)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
app43_in_ggga(x1, x2, x3, x4)  =  app43_in_ggga(x1, x2, x3)
app43_out_ggga(x1, x2, x3, x4)  =  app43_out_ggga(x1, x2, x3, x4)
U13_ggga(x1, x2, x3, x4, x5, x6)  =  U13_ggga(x1, x2, x3, x4, x6)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)

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

(215) UsableRulesProof (EQUIVALENT transformation)

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

(216) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
U2_GGAA(T62, T63, T64, X126, X127, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less14_in_gg(x1, x2)  =  less14_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less14_out_gg(x1, x2)  =  less14_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
PART24_IN_GGAA(x1, x2, x3, x4)  =  PART24_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)

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

(217) PiDPToQDPProof (SOUND transformation)

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

(218) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64)) → U2_GGAA(T62, T63, T64, less14_in_gg(T62, T63))
U2_GGAA(T62, T63, T64, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64)
PART24_IN_GGAA(T84, .(T85, T86)) → PART24_IN_GGAA(T84, T86)

The TRS R consists of the following rules:

less14_in_gg(0, s(T33)) → less14_out_gg(0, s(T33))
less14_in_gg(s(T38), s(T39)) → U1_gg(T38, T39, less14_in_gg(T38, T39))
U1_gg(T38, T39, less14_out_gg(T38, T39)) → less14_out_gg(s(T38), s(T39))

The set Q consists of the following terms:

less14_in_gg(x0, x1)
U1_gg(x0, x1, x2)

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

(219) 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:

• U2_GGAA(T62, T63, T64, less14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• PART24_IN_GGAA(T84, .(T85, T86)) → PART24_IN_GGAA(T84, T86)
  The graph contains the following edges 1 >= 1, 2 > 2

• PART24_IN_GGAA(T62, .(T63, T64)) → U2_GGAA(T62, T63, T64, less14_in_gg(T62, T63))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(220) YES