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

(0) Obligation:

Clauses:

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

Queries:

qs(a,g).

(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\nT2 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\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))) | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nqs(T1, T2) !=? qs(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 2 | ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 3 | ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 2\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 3 | ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(gt(T26, T27), ','(part(T26, T28, X47, X48), ','(qs(.(T27, X47), X9), ','(qs(X48, X10), app(X9, .(T26, X10), T8)))))\n\nT8 is ground\nX9 is free; \nX10 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: gt(T26, T27)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(part(T31, T32, X47, X48), ','(qs(.(T33, X47), X9), ','(qs(X48, X10), app(X9, .(T31, X10), T8))))\n\nT8 is ground\nT31 is ground\nT33 is ground\nX9 is free; \nX10 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: gt(T26, T27), SCOPE: 3, CLAUSE: 7 | gt(T26, T27), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: gt(T26, T27), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: gt(T26, T27), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: gt(T46, T47)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: part(T31, T32, X47, X48)\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(qs(.(T33, T53), X9), ','(qs(T54, X10), app(X9, .(T31, X10), T8)))\n\nT8 is ground\nT31 is ground\nT33 is ground\nT53 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 2 | part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 2\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(gt(T70, T73), part(T70, T74, X119, X120))\n\nT70 is ground\nX119 is free; \nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: gt(T70, T73)\n\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: part(T70, T77, X119, X120)\n\nT70 is ground\nX119 is free; \nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 3\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(le(T95, T98), part(T95, T99, X170, X171))\n\nT95 is ground\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T95, T98)\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T95, T102, X170, X171)\n\nT95 is ground\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: le(T95, T98), SCOPE: 5, CLAUSE: 9 | le(T95, T98), SCOPE: 5, CLAUSE: 10 | le(T95, T98), SCOPE: 5, CLAUSE: 11\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T95, T98), SCOPE: 5, CLAUSE: 9\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T95, T98), SCOPE: 5, CLAUSE: 10 | le(T95, T98), SCOPE: 5, CLAUSE: 11\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T113, T115)\n\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: le(T95, T98), SCOPE: 5, CLAUSE: 10\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T95, T98), SCOPE: 5, CLAUSE: 11\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: qs(.(T33, T53), X9)\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(qs(T54, X10), app(T131, .(T31, X10), T8))\n\nT8 is ground\nT31 is ground\nT131 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 0 | qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 1\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 0\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 1\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(part(T148, T149, X269, X270), ','(qs(X269, X271), ','(qs(X270, X272), app(X271, .(T148, X272), X273))))\n\nT148 is ground\nT149 is ground\nX273 is free; \nX269 is free; \nX270 is free; \nX271 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: part(T148, T149, X269, X270)\n\nT148 is ground\nT149 is ground\nX269 is free; \nX270 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(qs(T153, X271), ','(qs(T154, X272), app(X271, .(T148, X272), X273)))\n\nT148 is ground\nT153 is ground\nT154 is ground\nX273 is free; \nX271 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: qs(T153, X271)\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(qs(T154, X272), app(T160, .(T148, X272), X273))\n\nT148 is ground\nT154 is ground\nT160 is ground\nX273 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: qs(T153, X271), SCOPE: 7, CLAUSE: 0 | qs(T153, X271), SCOPE: 7, CLAUSE: 1\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: qs(T153, X271), SCOPE: 7, CLAUSE: 0\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: qs(T153, X271), SCOPE: 7, CLAUSE: 1\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(part(T171, T172, X331, X332), ','(qs(X331, X333), ','(qs(X332, X334), app(X333, .(T171, X334), X335))))\n\nT171 is ground\nT172 is ground\nX335 is free; \nX331 is free; \nX332 is free; \nX333 is free; \nX334 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: qs(T154, X272)\n\nT154 is ground\nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: app(T160, .(T148, T178), X273)\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 5 | app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 6\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 5\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 6\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(T202, .(T203, T204), X391)\n\nT202 is ground\nT203 is ground\nT204 is ground\nX391 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: qs(T54, X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: app(T131, .(T31, T217), T8)\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: qs(T54, X10), SCOPE: 9, CLAUSE: 0 | qs(T54, X10), SCOPE: 9, CLAUSE: 1\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: qs(T54, X10), SCOPE: 9, CLAUSE: 0\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: qs(T54, X10), SCOPE: 9, CLAUSE: 1\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(part(T230, T231, X448, X449), ','(qs(X448, X450), ','(qs(X449, X451), app(X450, .(T230, X451), X452))))\n\nX452 is free\nX448 is free; \nX449 is free; \nX450 is free; \nX451 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: part(T230, T231, X448, X449)\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(qs(T235, X450), ','(qs(T236, X451), app(X450, .(T237, X451), X452)))\n\nT235 is ground\nX452 is free; \nX450 is free; \nX451 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 2 | part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 3 | part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 4\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 2\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 3 | part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 4\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(gt(T256, T257), part(T256, T258, X503, X504))\n\nX503 is free\nX504 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: gt(T256, T257)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: part(T261, T262, X503, X504)\n\nT261 is ground\nX503 is free; \nX504 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 3\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 4\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(le(T283, T284), part(T283, T285, X554, X555))\n\nX554 is free\nX555 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: le(T283, T284)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: part(T288, T289, X554, X555)\n\nT288 is ground\nX554 is free; \nX555 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: le(T283, T284), SCOPE: 11, CLAUSE: 9 | le(T283, T284), SCOPE: 11, CLAUSE: 10 | le(T283, T284), SCOPE: 11, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: le(T283, T284), SCOPE: 11, CLAUSE: 9\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: le(T283, T284), SCOPE: 11, CLAUSE: 10 | le(T283, T284), SCOPE: 11, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: le(T302, T303)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: le(T283, T284), SCOPE: 11, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: le(T283, T284), SCOPE: 11, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: qs(T235, X450)\n\nT235 is ground\nX450 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(qs(T236, X451), app(T319, .(T237, X451), X452))\n\nT319 is ground\nX452 is free; \nX451 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: qs(T236, X451)\n\nX451 is free", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: app(T319, .(T325, T324), X452)\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 5 | app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 6\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 5\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 6\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: app(T349, .(T352, T353), X657)\n\nT349 is ground\nX657 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 5 | app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 6\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 5\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 6\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: app(T390, .(T391, T394), T393)\n\nT390 is ground\nT391 is ground\nT393 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: ','(le(T421, T422), ','(part(T421, T423, X740, X741), ','(qs(X740, X9), ','(qs(.(T422, X741), X10), app(X9, .(T421, X10), T8)))))\n\nT8 is ground\nX9 is free; \nX10 is free; \nX740 is free; \nX741 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: le(T421, T422)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: ','(part(T426, T427, X740, X741), ','(qs(X740, X9), ','(qs(.(T428, X741), X10), app(X9, .(T426, X10), T8))))\n\nT8 is ground\nT426 is ground\nX9 is free; \nX10 is free; \nX740 is free; \nX741 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: part(T426, T427, X740, X741)\n\nT426 is ground\nX740 is free; \nX741 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: ','(qs(T434, X9), ','(qs(.(T436, T435), X10), app(X9, .(T426, X10), T8)))\n\nT8 is ground\nT426 is ground\nT434 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: qs(T434, X9)\n\nT434 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: ','(qs(.(T436, T435), X10), app(T442, .(T426, X10), T8))\n\nT8 is ground\nT426 is ground\nT442 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: ','(qs([], X9), ','(qs([], X10), app(X9, .(T452, X10), T8)))\n\nT8 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: qs([], X9)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: ','(qs([], X10), app(T455, .(T452, X10), T8))\n\nT8 is ground\nT455 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: qs([], X9), SCOPE: 14, CLAUSE: 0 | qs([], X9), SCOPE: 14, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: qs([], X9), SCOPE: 14, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: qs([], X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: app(T455, .(T452, T460), T8)\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 5 | app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 6\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 5\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 6\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: app(T487, .(T491, T489), T490)\n\nT487 is ground\nT489 is ground\nT490 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 178 [arrowhead = none ];
178 -> 2;

179 [label="EVAL with clause\nqs(.(X4, X5), X6) :- ','(part(X4, X5, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(X4, X10), X6)))).\nand substitution\nX4 -> T9\nX5 -> T10\nT1 -> .(T9, T10)\nT2 -> T8\nX6 -> T8\nT6 -> T9\nT7 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 179 [arrowhead = none ];
179 -> 3;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 180 [arrowhead = none ];
180 -> 4;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 181 [arrowhead = none ];
181 -> 5;

182 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 182 [arrowhead = none ];
182 -> 175;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 183 [arrowhead = none ];
183 -> 176;

184 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 184 [arrowhead = none ];
184 -> 6;

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 185 [arrowhead = none ];
185 -> 7;

186 [label="EVAL with clause\npart(X42, .(X43, X44), .(X43, X45), X46) :- ','(gt(X42, X43), part(X42, X44, X45, X46)).\nand substitution\nT9 -> T26\nX42 -> T26\nX43 -> T27\nX44 -> T28\nT10 -> .(T27, T28)\nX45 -> X47\nX7 -> .(T27, X47)\nX8 -> X48\nX46 -> X48\nT23 -> T26\nT24 -> T27\nT25 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 186 [arrowhead = none ];
186 -> 8;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 187 [arrowhead = none ];
187 -> 9;

188 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 188 [arrowhead = none ];
188 -> 141;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 189 [arrowhead = none ];
189 -> 142;

190 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 190 [arrowhead = none ];
190 -> 10;

191 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nT33 is ground\nreplacements:\nT26 -> T31\nT28 -> T32\nT27 -> T33", fontsize=14, style = filled, fillcolor = violet];
8 -> 191 [arrowhead = none ];
191 -> 11;

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 192 [arrowhead = none ];
192 -> 12;

193 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 193 [arrowhead = none ];
193 -> 20;

194 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nreplacements:\nX47 -> T53\nX48 -> T54", fontsize=14, style = filled, fillcolor = violet];
11 -> 194 [arrowhead = none ];
194 -> 21;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 195 [arrowhead = none ];
195 -> 13;

196 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 196 [arrowhead = none ];
196 -> 14;

197 [label="EVAL with clause\ngt(s(X66), s(X67)) :- gt(X66, X67).\nand substitution\nX66 -> T46\nT26 -> s(T46)\nX67 -> T47\nT27 -> s(T47)\nT44 -> T46\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 197 [arrowhead = none ];
197 -> 15;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 198 [arrowhead = none ];
198 -> 16;

199 [label="EVAL with clause\ngt(s(0), 0).\nand substitution\nT26 -> s(0)\nT27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 199 [arrowhead = none ];
199 -> 17;

200 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 200 [arrowhead = none ];
200 -> 18;

201 [label="INSTANCE with matching:\nT26 -> T46\nT27 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 201 [arrowhead = none , style=dashed];
201 -> 10[style=dashed];

202 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 202 [arrowhead = none ];
202 -> 19;

203 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 203 [arrowhead = none ];
203 -> 22;

204 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 204 [arrowhead = none ];
204 -> 51;

205 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nreplacements:\nX9 -> T131", fontsize=14, style = filled, fillcolor = violet];
21 -> 205 [arrowhead = none ];
205 -> 52;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 206 [arrowhead = none ];
206 -> 23;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 207 [arrowhead = none ];
207 -> 24;

208 [label="EVAL with clause\npart(X114, .(X115, X116), .(X115, X117), X118) :- ','(gt(X114, X115), part(X114, X116, X117, X118)).\nand substitution\nT31 -> T70\nX114 -> T70\nX115 -> T73\nX116 -> T74\nT32 -> .(T73, T74)\nX117 -> X119\nX47 -> .(T73, X119)\nX48 -> X120\nX118 -> X120\nT71 -> T73\nT72 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 208 [arrowhead = none ];
208 -> 25;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 209 [arrowhead = none ];
209 -> 26;

210 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 210 [arrowhead = none ];
210 -> 29;

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 211 [arrowhead = none ];
211 -> 30;

212 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 212 [arrowhead = none ];
212 -> 27;

213 [label="SPLIT 2\nnew knowledge:\nT73 is ground\nreplacements:\nT74 -> T77", fontsize=14, style = filled, fillcolor = violet];
25 -> 213 [arrowhead = none ];
213 -> 28;

214 [label="INSTANCE with matching:\nT26 -> T70\nT27 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 214 [arrowhead = none , style=dashed];
214 -> 10[style=dashed];

215 [label="INSTANCE with matching:\nT31 -> T70\nT32 -> T77\nX47 -> X119\nX48 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 215 [arrowhead = none , style=dashed];
215 -> 20[style=dashed];

216 [label="EVAL with clause\npart(X165, .(X166, X167), X168, .(X166, X169)) :- ','(le(X165, X166), part(X165, X167, X168, X169)).\nand substitution\nT31 -> T95\nX165 -> T95\nX166 -> T98\nX167 -> T99\nT32 -> .(T98, T99)\nX47 -> X170\nX168 -> X170\nX169 -> X171\nX48 -> .(T98, X171)\nT96 -> T98\nT97 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 216 [arrowhead = none ];
216 -> 31;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 217 [arrowhead = none ];
217 -> 32;

218 [label="EVAL with clause\npart(X210, [], [], []).\nand substitution\nT31 -> T128\nX210 -> T128\nT32 -> []\nX47 -> []\nX48 -> []", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 218 [arrowhead = none ];
218 -> 48;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 219 [arrowhead = none ];
219 -> 49;

220 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 220 [arrowhead = none ];
220 -> 33;

221 [label="SPLIT 2\nreplacements:\nT99 -> T102", fontsize=14, style = filled, fillcolor = violet];
31 -> 221 [arrowhead = none ];
221 -> 34;

222 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 222 [arrowhead = none ];
222 -> 35;

223 [label="INSTANCE with matching:\nT31 -> T95\nT32 -> T102\nX47 -> X170\nX48 -> X171", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 223 [arrowhead = none , style=dashed];
223 -> 20[style=dashed];

224 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 224 [arrowhead = none ];
224 -> 36;

225 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 225 [arrowhead = none ];
225 -> 37;

226 [label="EVAL with clause\nle(s(X189), s(X190)) :- le(X189, X190).\nand substitution\nX189 -> T113\nT95 -> s(T113)\nX190 -> T115\nT98 -> s(T115)\nT114 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 226 [arrowhead = none ];
226 -> 38;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 227 [arrowhead = none ];
227 -> 39;

228 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 228 [arrowhead = none ];
228 -> 40;

229 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 229 [arrowhead = none ];
229 -> 41;

230 [label="INSTANCE with matching:\nT95 -> T113\nT98 -> T115", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 230 [arrowhead = none , style=dashed];
230 -> 33[style=dashed];

231 [label="EVAL with clause\nle(0, s(X199)).\nand substitution\nT95 -> 0\nX199 -> T122\nT98 -> s(T122)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 231 [arrowhead = none ];
231 -> 42;

232 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 232 [arrowhead = none ];
232 -> 43;

233 [label="EVAL with clause\nle(0, 0).\nand substitution\nT95 -> 0\nT98 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 233 [arrowhead = none ];
233 -> 45;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 234 [arrowhead = none ];
234 -> 46;

235 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 235 [arrowhead = none ];
235 -> 44;

236 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 236 [arrowhead = none ];
236 -> 47;

237 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 237 [arrowhead = none ];
237 -> 50;

238 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 238 [arrowhead = none ];
238 -> 53;

239 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 239 [arrowhead = none ];
239 -> 80;

240 [label="SPLIT 2\nreplacements:\nX10 -> T217", fontsize=14, style = filled, fillcolor = violet];
52 -> 240 [arrowhead = none ];
240 -> 81;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 241 [arrowhead = none ];
241 -> 54;

242 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 242 [arrowhead = none ];
242 -> 55;

243 [label="ONLY EVAL with clause\nqs(.(X266, X267), X268) :- ','(part(X266, X267, X269, X270), ','(qs(X269, X271), ','(qs(X270, X272), app(X271, .(X266, X272), X268)))).\nand substitution\nT33 -> T148\nX266 -> T148\nT53 -> T149\nX267 -> T149\nX9 -> X273\nX268 -> X273", fontsize=14, style = filled, fillcolor = white];
54 -> 243 [arrowhead = none ];
243 -> 56;

244 [label="BACKTRACK\nfor clause: qs([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 244 [arrowhead = none ];
244 -> 79;

245 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 245 [arrowhead = none ];
245 -> 57;

246 [label="SPLIT 2\nnew knowledge:\nT153 is ground\nT154 is ground\nreplacements:\nX269 -> T153\nX270 -> T154", fontsize=14, style = filled, fillcolor = violet];
56 -> 246 [arrowhead = none ];
246 -> 58;

247 [label="INSTANCE with matching:\nT31 -> T148\nT32 -> T149\nX47 -> X269\nX48 -> X270", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 247 [arrowhead = none , style=dashed];
247 -> 20[style=dashed];

248 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 248 [arrowhead = none ];
248 -> 59;

249 [label="SPLIT 2\nnew knowledge:\nT160 is ground\nreplacements:\nX271 -> T160", fontsize=14, style = filled, fillcolor = violet];
58 -> 249 [arrowhead = none ];
249 -> 60;

250 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 250 [arrowhead = none ];
250 -> 61;

251 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
60 -> 251 [arrowhead = none ];
251 -> 69;

252 [label="SPLIT 2\nnew knowledge:\nT178 is ground\nreplacements:\nX272 -> T178", fontsize=14, style = filled, fillcolor = violet];
60 -> 252 [arrowhead = none ];
252 -> 70;

253 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 253 [arrowhead = none ];
253 -> 62;

254 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 254 [arrowhead = none ];
254 -> 63;

255 [label="EVAL with clause\nqs(.(X328, X329), X330) :- ','(part(X328, X329, X331, X332), ','(qs(X331, X333), ','(qs(X332, X334), app(X333, .(X328, X334), X330)))).\nand substitution\nX328 -> T171\nX329 -> T172\nT153 -> .(T171, T172)\nX271 -> X335\nX330 -> X335", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 255 [arrowhead = none ];
255 -> 64;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 256 [arrowhead = none ];
256 -> 65;

257 [label="EVAL with clause\nqs([], []).\nand substitution\nT153 -> []\nX271 -> []", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 257 [arrowhead = none ];
257 -> 66;

258 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 258 [arrowhead = none ];
258 -> 67;

259 [label="INSTANCE with matching:\nT148 -> T171\nT149 -> T172\nX269 -> X331\nX270 -> X332\nX271 -> X333\nX272 -> X334\nX273 -> X335", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 259 [arrowhead = none , style=dashed];
259 -> 56[style=dashed];

260 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 260 [arrowhead = none ];
260 -> 68;

261 [label="INSTANCE with matching:\nT153 -> T154\nX271 -> X272", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 261 [arrowhead = none , style=dashed];
261 -> 59[style=dashed];

262 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 262 [arrowhead = none ];
262 -> 71;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 263 [arrowhead = none ];
263 -> 72;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 264 [arrowhead = none ];
264 -> 73;

265 [label="EVAL with clause\napp(.(X387, X388), X389, .(X387, X390)) :- app(X388, X389, X390).\nand substitution\nX387 -> T201\nX388 -> T202\nT160 -> .(T201, T202)\nT148 -> T203\nT178 -> T204\nX389 -> .(T203, T204)\nX390 -> X391\nX273 -> .(T201, X391)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 265 [arrowhead = none ];
265 -> 74;

266 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 266 [arrowhead = none ];
266 -> 75;

267 [label="EVAL with clause\napp([], X403, X403).\nand substitution\nT160 -> []\nT148 -> T213\nT178 -> T214\nX403 -> .(T213, T214)\nX273 -> .(T213, T214)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 267 [arrowhead = none ];
267 -> 76;

268 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 268 [arrowhead = none ];
268 -> 77;

269 [label="INSTANCE with matching:\nT160 -> T202\nT148 -> T203\nT178 -> T204\nX273 -> X391", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 269 [arrowhead = none , style=dashed];
269 -> 70[style=dashed];

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 270 [arrowhead = none ];
270 -> 78;

271 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 271 [arrowhead = none ];
271 -> 82;

272 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 272 [arrowhead = none ];
272 -> 133;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 273 [arrowhead = none ];
273 -> 83;

274 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 274 [arrowhead = none ];
274 -> 84;

275 [label="EVAL with clause\nqs(.(X445, X446), X447) :- ','(part(X445, X446, X448, X449), ','(qs(X448, X450), ','(qs(X449, X451), app(X450, .(X445, X451), X447)))).\nand substitution\nX445 -> T230\nX446 -> T231\nT54 -> .(T230, T231)\nX10 -> X452\nX447 -> X452\nT228 -> T230\nT229 -> T231", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 275 [arrowhead = none ];
275 -> 85;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 276 [arrowhead = none ];
276 -> 86;

277 [label="EVAL with clause\nqs([], []).\nand substitution\nT54 -> []\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 277 [arrowhead = none ];
277 -> 130;

278 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 278 [arrowhead = none ];
278 -> 131;

279 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 279 [arrowhead = none ];
279 -> 87;

280 [label="SPLIT 2\nnew knowledge:\nT235 is ground\nreplacements:\nX448 -> T235\nX449 -> T236\nT230 -> T237", fontsize=14, style = filled, fillcolor = violet];
85 -> 280 [arrowhead = none ];
280 -> 88;

281 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 281 [arrowhead = none ];
281 -> 89;

282 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
88 -> 282 [arrowhead = none ];
282 -> 118;

283 [label="SPLIT 2\nnew knowledge:\nT319 is ground\nreplacements:\nX450 -> T319", fontsize=14, style = filled, fillcolor = violet];
88 -> 283 [arrowhead = none ];
283 -> 119;

284 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
89 -> 284 [arrowhead = none ];
284 -> 90;

285 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
89 -> 285 [arrowhead = none ];
285 -> 91;

286 [label="EVAL with clause\npart(X498, .(X499, X500), .(X499, X501), X502) :- ','(gt(X498, X499), part(X498, X500, X501, X502)).\nand substitution\nT230 -> T256\nX498 -> T256\nX499 -> T257\nX500 -> T258\nT231 -> .(T257, T258)\nX501 -> X503\nX448 -> .(T257, X503)\nX449 -> X504\nX502 -> X504\nT253 -> T256\nT254 -> T257\nT255 -> T258", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 286 [arrowhead = none ];
286 -> 92;

287 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 287 [arrowhead = none ];
287 -> 93;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 288 [arrowhead = none ];
288 -> 96;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 289 [arrowhead = none ];
289 -> 97;

290 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 290 [arrowhead = none ];
290 -> 94;

291 [label="SPLIT 2\nnew knowledge:\nT261 is ground\nT257 is ground\nreplacements:\nT256 -> T261\nT258 -> T262", fontsize=14, style = filled, fillcolor = violet];
92 -> 291 [arrowhead = none ];
291 -> 95;

292 [label="INSTANCE with matching:\nT26 -> T256\nT27 -> T257", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 292 [arrowhead = none , style=dashed];
292 -> 10[style=dashed];

293 [label="INSTANCE with matching:\nT31 -> T261\nT32 -> T262\nX47 -> X503\nX48 -> X504", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 293 [arrowhead = none , style=dashed];
293 -> 20[style=dashed];

294 [label="EVAL with clause\npart(X549, .(X550, X551), X552, .(X550, X553)) :- ','(le(X549, X550), part(X549, X551, X552, X553)).\nand substitution\nT230 -> T283\nX549 -> T283\nX550 -> T284\nX551 -> T285\nT231 -> .(T284, T285)\nX448 -> X554\nX552 -> X554\nX553 -> X555\nX449 -> .(T284, X555)\nT280 -> T283\nT281 -> T284\nT282 -> T285", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 294 [arrowhead = none ];
294 -> 98;

295 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 295 [arrowhead = none ];
295 -> 99;

296 [label="EVAL with clause\npart(X594, [], [], []).\nand substitution\nT230 -> T316\nX594 -> T316\nT231 -> []\nX448 -> []\nX449 -> []", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 296 [arrowhead = none ];
296 -> 115;

297 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 297 [arrowhead = none ];
297 -> 116;

298 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
98 -> 298 [arrowhead = none ];
298 -> 100;

299 [label="SPLIT 2\nnew knowledge:\nT288 is ground\nreplacements:\nT283 -> T288\nT285 -> T289", fontsize=14, style = filled, fillcolor = violet];
98 -> 299 [arrowhead = none ];
299 -> 101;

300 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
100 -> 300 [arrowhead = none ];
300 -> 102;

301 [label="INSTANCE with matching:\nT31 -> T288\nT32 -> T289\nX47 -> X554\nX48 -> X555", fontsize=14, style = filled, fillcolor = lightgrey];
101 -> 301 [arrowhead = none , style=dashed];
301 -> 20[style=dashed];

302 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 302 [arrowhead = none ];
302 -> 103;

303 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 303 [arrowhead = none ];
303 -> 104;

304 [label="EVAL with clause\nle(s(X573), s(X574)) :- le(X573, X574).\nand substitution\nX573 -> T302\nT283 -> s(T302)\nX574 -> T303\nT284 -> s(T303)\nT300 -> T302\nT301 -> T303", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 304 [arrowhead = none ];
304 -> 105;

305 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 305 [arrowhead = none ];
305 -> 106;

306 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 306 [arrowhead = none ];
306 -> 107;

307 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 307 [arrowhead = none ];
307 -> 108;

308 [label="INSTANCE with matching:\nT283 -> T302\nT284 -> T303", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 308 [arrowhead = none , style=dashed];
308 -> 100[style=dashed];

309 [label="EVAL with clause\nle(0, s(X583)).\nand substitution\nT283 -> 0\nX583 -> T310\nT284 -> s(T310)", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 309 [arrowhead = none ];
309 -> 109;

310 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 310 [arrowhead = none ];
310 -> 110;

311 [label="EVAL with clause\nle(0, 0).\nand substitution\nT283 -> 0\nT284 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 311 [arrowhead = none ];
311 -> 112;

312 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 312 [arrowhead = none ];
312 -> 113;

313 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 313 [arrowhead = none ];
313 -> 111;

314 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
112 -> 314 [arrowhead = none ];
314 -> 114;

315 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 315 [arrowhead = none ];
315 -> 117;

316 [label="INSTANCE with matching:\nT54 -> T235\nX10 -> X450", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 316 [arrowhead = none , style=dashed];
316 -> 80[style=dashed];

317 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
119 -> 317 [arrowhead = none ];
317 -> 120;

318 [label="SPLIT 2\nreplacements:\nX451 -> T324\nT237 -> T325", fontsize=14, style = filled, fillcolor = violet];
119 -> 318 [arrowhead = none ];
318 -> 121;

319 [label="INSTANCE with matching:\nT54 -> T236\nX10 -> X451", fontsize=14, style = filled, fillcolor = lightgrey];
120 -> 319 [arrowhead = none , style=dashed];
319 -> 80[style=dashed];

320 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
121 -> 320 [arrowhead = none ];
320 -> 122;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 321 [arrowhead = none ];
321 -> 123;

322 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 322 [arrowhead = none ];
322 -> 124;

323 [label="EVAL with clause\napp(.(X653, X654), X655, .(X653, X656)) :- app(X654, X655, X656).\nand substitution\nX653 -> T348\nX654 -> T349\nT319 -> .(T348, T349)\nT325 -> T352\nT324 -> T353\nX655 -> .(T352, T353)\nX656 -> X657\nX452 -> .(T348, X657)\nT350 -> T352\nT351 -> T353", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 323 [arrowhead = none ];
323 -> 125;

324 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 324 [arrowhead = none ];
324 -> 126;

325 [label="EVAL with clause\napp([], X669, X669).\nand substitution\nT319 -> []\nT325 -> T362\nT324 -> T363\nX669 -> .(T362, T363)\nX452 -> .(T362, T363)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 325 [arrowhead = none ];
325 -> 127;

326 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 326 [arrowhead = none ];
326 -> 128;

327 [label="INSTANCE with matching:\nT319 -> T349\nT325 -> T352\nT324 -> T353\nX452 -> X657", fontsize=14, style = filled, fillcolor = lightgrey];
125 -> 327 [arrowhead = none , style=dashed];
327 -> 121[style=dashed];

328 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
127 -> 328 [arrowhead = none ];
328 -> 129;

329 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 329 [arrowhead = none ];
329 -> 132;

330 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
133 -> 330 [arrowhead = none ];
330 -> 134;

331 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
133 -> 331 [arrowhead = none ];
331 -> 135;

332 [label="EVAL with clause\napp(.(X691, X692), X693, .(X691, X694)) :- app(X692, X693, X694).\nand substitution\nX691 -> T389\nX692 -> T390\nT131 -> .(T389, T390)\nT31 -> T391\nT217 -> T394\nX693 -> .(T391, T394)\nX694 -> T393\nT8 -> .(T389, T393)\nT392 -> T394", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 332 [arrowhead = none ];
332 -> 136;

333 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 333 [arrowhead = none ];
333 -> 137;

334 [label="EVAL with clause\napp([], X705, X705).\nand substitution\nT131 -> []\nT31 -> T404\nT217 -> T405\nX705 -> .(T404, T405)\nT8 -> .(T404, T405)", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 334 [arrowhead = none ];
334 -> 138;

335 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 335 [arrowhead = none ];
335 -> 139;

336 [label="INSTANCE with matching:\nT131 -> T390\nT31 -> T391\nT217 -> T394\nT8 -> T393", fontsize=14, style = filled, fillcolor = lightgrey];
136 -> 336 [arrowhead = none , style=dashed];
336 -> 81[style=dashed];

337 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 337 [arrowhead = none ];
337 -> 140;

338 [label="EVAL with clause\npart(X735, .(X736, X737), X738, .(X736, X739)) :- ','(le(X735, X736), part(X735, X737, X738, X739)).\nand substitution\nT9 -> T421\nX735 -> T421\nX736 -> T422\nX737 -> T423\nT10 -> .(T422, T423)\nX7 -> X740\nX738 -> X740\nX739 -> X741\nX8 -> .(T422, X741)\nT418 -> T421\nT419 -> T422\nT420 -> T423", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 338 [arrowhead = none ];
338 -> 143;

339 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 339 [arrowhead = none ];
339 -> 144;

340 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 340 [arrowhead = none ];
340 -> 151;

341 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 341 [arrowhead = none ];
341 -> 152;

342 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
143 -> 342 [arrowhead = none ];
342 -> 145;

343 [label="SPLIT 2\nnew knowledge:\nT426 is ground\nreplacements:\nT421 -> T426\nT423 -> T427\nT422 -> T428", fontsize=14, style = filled, fillcolor = violet];
143 -> 343 [arrowhead = none ];
343 -> 146;

344 [label="INSTANCE with matching:\nT283 -> T421\nT284 -> T422", fontsize=14, style = filled, fillcolor = lightgrey];
145 -> 344 [arrowhead = none , style=dashed];
344 -> 100[style=dashed];

345 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
146 -> 345 [arrowhead = none ];
345 -> 147;

346 [label="SPLIT 2\nnew knowledge:\nT434 is ground\nreplacements:\nX740 -> T434\nX741 -> T435\nT428 -> T436", fontsize=14, style = filled, fillcolor = violet];
146 -> 346 [arrowhead = none ];
346 -> 148;

347 [label="INSTANCE with matching:\nT31 -> T426\nT32 -> T427\nX47 -> X740\nX48 -> X741", fontsize=14, style = filled, fillcolor = lightgrey];
147 -> 347 [arrowhead = none , style=dashed];
347 -> 20[style=dashed];

348 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
148 -> 348 [arrowhead = none ];
348 -> 149;

349 [label="SPLIT 2\nnew knowledge:\nT442 is ground\nreplacements:\nX9 -> T442", fontsize=14, style = filled, fillcolor = violet];
148 -> 349 [arrowhead = none ];
349 -> 150;

350 [label="INSTANCE with matching:\nT54 -> T434\nX10 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
149 -> 350 [arrowhead = none , style=dashed];
350 -> 80[style=dashed];

351 [label="INSTANCE with matching:\nT54 -> .(T436, T435)\nT131 -> T442\nT31 -> T426", fontsize=14, style = filled, fillcolor = lightgrey];
150 -> 351 [arrowhead = none , style=dashed];
351 -> 52[style=dashed];

352 [label="EVAL with clause\npart(X793, [], [], []).\nand substitution\nT9 -> T452\nX793 -> T452\nT10 -> []\nX7 -> []\nX8 -> []\nT451 -> T452", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 352 [arrowhead = none ];
352 -> 153;

353 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 353 [arrowhead = none ];
353 -> 154;

354 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 354 [arrowhead = none ];
354 -> 171;

355 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
153 -> 355 [arrowhead = none ];
355 -> 155;

356 [label="SPLIT 2\nnew knowledge:\nT455 is ground\nreplacements:\nX9 -> T455", fontsize=14, style = filled, fillcolor = violet];
153 -> 356 [arrowhead = none ];
356 -> 156;

357 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
155 -> 357 [arrowhead = none ];
357 -> 157;

358 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
156 -> 358 [arrowhead = none ];
358 -> 161;

359 [label="SPLIT 2\nnew knowledge:\nT460 is ground\nreplacements:\nX10 -> T460", fontsize=14, style = filled, fillcolor = violet];
156 -> 359 [arrowhead = none ];
359 -> 162;

360 [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];
157 -> 360 [arrowhead = none ];
360 -> 158;

361 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX9 -> []", fontsize=14, style = filled, fillcolor = white];
158 -> 361 [arrowhead = none ];
361 -> 159;

362 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
159 -> 362 [arrowhead = none ];
362 -> 160;

363 [label="INSTANCE with matching:\nX9 -> X10", fontsize=14, style = filled, fillcolor = lightgrey];
161 -> 363 [arrowhead = none , style=dashed];
363 -> 155[style=dashed];

364 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
162 -> 364 [arrowhead = none ];
364 -> 163;

365 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
163 -> 365 [arrowhead = none ];
365 -> 164;

366 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
163 -> 366 [arrowhead = none ];
366 -> 165;

367 [label="EVAL with clause\napp(.(X865, X866), X867, .(X865, X868)) :- app(X866, X867, X868).\nand substitution\nX865 -> T486\nX866 -> T487\nT455 -> .(T486, T487)\nT452 -> T491\nT460 -> T489\nX867 -> .(T491, T489)\nX868 -> T490\nT8 -> .(T486, T490)\nT488 -> T491", fontsize=14, style = filled, fillcolor = lightblue];
164 -> 367 [arrowhead = none ];
367 -> 166;

368 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
164 -> 368 [arrowhead = none ];
368 -> 167;

369 [label="EVAL with clause\napp([], X879, X879).\nand substitution\nT455 -> []\nT452 -> T501\nT460 -> T502\nX879 -> .(T501, T502)\nT8 -> .(T501, T502)", fontsize=14, style = filled, fillcolor = lightblue];
165 -> 369 [arrowhead = none ];
369 -> 168;

370 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
165 -> 370 [arrowhead = none ];
370 -> 169;

371 [label="INSTANCE with matching:\nT455 -> T487\nT452 -> T491\nT460 -> T489\nT8 -> T490", fontsize=14, style = filled, fillcolor = lightgrey];
166 -> 371 [arrowhead = none , style=dashed];
371 -> 162[style=dashed];

372 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
168 -> 372 [arrowhead = none ];
372 -> 170;

373 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
171 -> 373 [arrowhead = none ];
373 -> 172;

374 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
171 -> 374 [arrowhead = none ];
374 -> 173;

375 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
172 -> 375 [arrowhead = none ];
375 -> 174;

376 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
175 -> 376 [arrowhead = none ];
376 -> 177;

}

(2) Obligation:

Clauses:

gt10(s(T46), s(T47)) :- gt10(T46, T47).
gt10(s(0), 0).
part20(T70, .(T73, T74), .(T73, X119), X120) :- gt10(T70, T73).
part20(T70, .(T73, T77), .(T73, X119), X120) :- ','(gt10(T70, T73), part20(T70, T77, X119, X120)).
part20(T95, .(T98, T99), X170, .(T98, X171)) :- le33(T95, T98).
part20(T95, .(T98, T102), X170, .(T98, X171)) :- ','(le33(T95, T98), part20(T95, T102, X170, X171)).
part20(T128, [], [], []).
le33(s(T113), s(T115)) :- le33(T113, T115).
le33(0, s(T122)).
le33(0, 0).
p56(T148, T149, X269, X270, X271, X272, X273) :- part20(T148, T149, X269, X270).
p56(T148, T149, T153, T154, X271, X272, X273) :- ','(part20(T148, T149, T153, T154), qs59(T153, X271)).
p56(T148, T149, T153, T154, T160, X272, X273) :- ','(part20(T148, T149, T153, T154), ','(qs59(T153, T160), qs59(T154, X272))).
p56(T148, T149, T153, T154, T160, T178, X273) :- ','(part20(T148, T149, T153, T154), ','(qs59(T153, T160), ','(qs59(T154, T178), app70(T160, T148, T178, X273)))).
qs59(.(T171, T172), X335) :- p56(T171, T172, X331, X332, X333, X334, X335).
qs59([], []).
app70(.(T201, T202), T203, T204, .(T201, X391)) :- app70(T202, T203, T204, X391).
app70([], T213, T214, .(T213, T214)).
le100(s(T302), s(T303)) :- le100(T302, T303).
le100(0, s(T310)).
le100(0, 0).
qs80(.(T230, T231), X452) :- part87(T230, T231, X448, X449).
qs80(.(T237, T231), X452) :- ','(part87(T237, T231, T235, T236), qs80(T235, X450)).
qs80(.(T237, T231), X452) :- ','(part87(T237, T231, T235, T236), ','(qs80(T235, T319), qs80(T236, X451))).
qs80(.(T325, T231), X452) :- ','(part87(T325, T231, T235, T236), ','(qs80(T235, T319), ','(qs80(T236, T324), app121(T319, T325, T324, X452)))).
qs80([], []).
app121(.(T348, T349), T352, T353, .(T348, X657)) :- app121(T349, T352, T353, X657).
app121([], T362, T363, .(T362, T363)).
app81(.(T389, T390), T391, T394, .(T389, T393)) :- app81(T390, T391, T394, T393).
app81([], T404, T405, .(T404, T405)).
p52(T54, X10, T131, T31, T8) :- qs80(T54, X10).
p52(T54, T217, T131, T31, T8) :- ','(qs80(T54, T217), app81(T131, T31, T217, T8)).
qs155([]).
app162(.(T486, T487), T491, T489, .(T486, T490)) :- app162(T487, T491, T489, T490).
app162([], T501, T502, .(T501, T502)).
qs51(T148, T149, X273) :- p56(T148, T149, X269, X270, X271, X272, X273).
part87(T256, .(T257, T258), .(T257, X503), X504) :- gt10(T256, T257).
part87(T261, .(T257, T262), .(T257, X503), X504) :- ','(gt10(T261, T257), part20(T261, T262, X503, X504)).
part87(T283, .(T284, T285), X554, .(T284, X555)) :- le100(T283, T284).
part87(T288, .(T284, T289), X554, .(T284, X555)) :- ','(le100(T288, T284), part20(T288, T289, X554, X555)).
part87(T316, [], [], []).
qs1(.(T26, .(T27, T28)), T8) :- gt10(T26, T27).
qs1(.(T31, .(T33, T32)), T8) :- ','(gt10(T31, T33), part20(T31, T32, X47, X48)).
qs1(.(T31, .(T33, T32)), T8) :- ','(gt10(T31, T33), ','(part20(T31, T32, T53, T54), qs51(T33, T53, X9))).
qs1(.(T31, .(T33, T32)), T8) :- ','(gt10(T31, T33), ','(part20(T31, T32, T53, T54), ','(qs51(T33, T53, T131), p52(T54, X10, T131, T31, T8)))).
qs1(.(T421, .(T422, T423)), T8) :- le100(T421, T422).
qs1(.(T426, .(T428, T427)), T8) :- ','(le100(T426, T428), part20(T426, T427, X740, X741)).
qs1(.(T426, .(T436, T427)), T8) :- ','(le100(T426, T436), ','(part20(T426, T427, T434, T435), qs80(T434, X9))).
qs1(.(T426, .(T436, T427)), T8) :- ','(le100(T426, T436), ','(part20(T426, T427, T434, T435), ','(qs80(T434, T442), p52(.(T436, T435), X10, T442, T426, T8)))).
qs1(.(T452, []), T8) :- qs155(X9).
qs1(.(T452, []), T8) :- ','(qs155(T455), qs155(X10)).
qs1(.(T452, []), T8) :- ','(qs155(T455), ','(qs155(T460), app162(T455, T452, T460, T8))).
qs1([], []).
qs1([], []).

Queries:

qs1(a,g).

(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: (f,b)
gt10_in: (f,f) (b,f)
part20_in: (b,f,f,f) (f,f,f,f)
le33_in: (b,f) (f,f)
qs51_in: (b,f,f)
p56_in: (b,f,f,f,f,f,f) (f,f,f,f,f,f,f)
qs59_in: (f,f)
app70_in: (f,f,f,f) (f,b,f,f)
p52_in: (f,f,f,b,b)
qs80_in: (f,f)
part87_in: (f,f,f,f)
le100_in: (f,f)
app121_in: (f,f,f,f)
app81_in: (f,b,f,b)
app162_in: (b,f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)

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_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)

(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_AG(.(T26, .(T27, T28)), T8) → U44_AG(T26, T27, T28, T8, gt10_in_aa(T26, T27))
QS1_IN_AG(.(T26, .(T27, T28)), T8) → GT10_IN_AA(T26, T27)
GT10_IN_AA(s(T46), s(T47)) → U1_AA(T46, T47, gt10_in_aa(T46, T47))
GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)
QS1_IN_AG(.(T31, .(T33, T32)), T8) → U45_AG(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → PART20_IN_GAAA(T31, T32, X47, X48)
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GAAA(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GA(T70, T73)
GT10_IN_GA(s(T46), s(T47)) → U1_GA(T46, T47, gt10_in_ga(T46, T47))
GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)
PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GAAA(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GA(T95, T98)
LE33_IN_GA(s(T113), s(T115)) → U8_GA(T113, T115, le33_in_ga(T113, T115))
LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_GAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_AG(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → QS51_IN_GAA(T33, T53, X9)
QS51_IN_GAA(T148, T149, X273) → U37_GAA(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
QS51_IN_GAA(T148, T149, X273) → P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273)
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_GAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_GAAA(T148, T149, X269, X270)
P56_IN_GAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → U19_AA(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_AAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_AAAA(T148, T149, X269, X270)
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_AAAA(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_AA(T70, T73)
PART20_IN_AAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_AAAA(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_AAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_AAAA(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_AA(T95, T98)
LE33_IN_AA(s(T113), s(T115)) → U8_AA(T113, T115, le33_in_aa(T113, T115))
LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)
PART20_IN_AAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_AAAA(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_AAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AAAA(T160, T148, T178, X273)
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AAAA(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AGAA(T160, T148, T178, X273)
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AGAA(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_AG(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_AG(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → P52_IN_AAAGG(T54, X10, T131, T31, T8)
P52_IN_AAAGG(T54, X10, T131, T31, T8) → U33_AAAGG(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
P52_IN_AAAGG(T54, X10, T131, T31, T8) → QS80_IN_AA(T54, X10)
QS80_IN_AA(.(T230, T231), X452) → U22_AA(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
QS80_IN_AA(.(T230, T231), X452) → PART87_IN_AAAA(T230, T231, X448, X449)
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → U38_AAAA(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → GT10_IN_AA(T256, T257)
PART87_IN_AAAA(T261, .(T257, T262), .(T257, X503), X504) → U39_AAAA(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_AAAA(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → PART20_IN_GAAA(T261, T262, X503, X504)
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → U41_AAAA(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → LE100_IN_AA(T283, T284)
LE100_IN_AA(s(T302), s(T303)) → U21_AA(T302, T303, le100_in_aa(T302, T303))
LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)
PART87_IN_AAAA(T288, .(T284, T289), X554, .(T284, X555)) → U42_AAAA(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_AAAA(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → PART20_IN_GAAA(T288, T289, X554, X555)
QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_AA(T237, T231, X452, qs80_in_aa(T235, X450))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_AA(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_AA(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → APP121_IN_AAAA(T319, T325, T324, X452)
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → U31_AAAA(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_AA(T237, T231, X452, qs80_in_aa(T236, X451))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)
P52_IN_AAAGG(T54, T217, T131, T31, T8) → U34_AAAGG(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_AAAGG(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → APP81_IN_AGAG(T131, T31, T217, T8)
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → U32_AGAG(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)
QS1_IN_AG(.(T421, .(T422, T423)), T8) → U51_AG(T421, T422, T423, T8, le100_in_aa(T421, T422))
QS1_IN_AG(.(T421, .(T422, T423)), T8) → LE100_IN_AA(T421, T422)
QS1_IN_AG(.(T426, .(T428, T427)), T8) → U52_AG(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_AG(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → PART20_IN_GAAA(T426, T427, X740, X741)
QS1_IN_AG(.(T426, .(T436, T427)), T8) → U54_AG(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_AG(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → PART20_IN_GAAA(T426, T427, T434, T435)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_AG(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → QS80_IN_AA(T434, X9)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_AG(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_AG(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → P52_IN_AAAGG(.(T436, T435), X10, T442, T426, T8)
QS1_IN_AG(.(T452, []), T8) → U59_AG(T452, T8, qs155_in_a(X9))
QS1_IN_AG(.(T452, []), T8) → QS155_IN_A(X9)
QS1_IN_AG(.(T452, []), T8) → U60_AG(T452, T8, qs155_in_a(T455))
U60_AG(T452, T8, qs155_out_a(T455)) → U61_AG(T452, T8, qs155_in_a(X10))
U60_AG(T452, T8, qs155_out_a(T455)) → QS155_IN_A(X10)
U60_AG(T452, T8, qs155_out_a(T455)) → U62_AG(T452, T8, T455, qs155_in_a(T460))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → U63_AG(T452, T8, app162_in_gagg(T455, T452, T460, T8))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → APP162_IN_GAGG(T455, T452, T460, T8)
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → U36_GAGG(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
QS1_IN_AG(x1, x2)  =  QS1_IN_AG(x2)
U44_AG(x1, x2, x3, x4, x5)  =  U44_AG(x4, x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U45_AG(x1, x2, x3, x4, x5)  =  U45_AG(x4, x5)
U46_AG(x1, x2, x3, x4, x5)  =  U46_AG(x4, x5)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
GT10_IN_GA(x1, x2)  =  GT10_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)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)
U47_AG(x1, x2, x3, x4, x5)  =  U47_AG(x1, x2, x4, x5)
U48_AG(x1, x2, x3, x4, x5)  =  U48_AG(x4, x5)
QS51_IN_GAA(x1, x2, x3)  =  QS51_IN_GAA(x1)
U37_GAA(x1, x2, x3, x4)  =  U37_GAA(x1, x4)
P56_IN_GAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_GAAAAAA(x1)
U9_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_GAAAAAA(x1, x8)
U10_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GAAAAAA(x1, x8)
U11_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GAAAAAA(x1, x8)
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U9_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AAAAAAA(x8)
PART20_IN_AAAA(x1, x2, x3, x4)  =  PART20_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)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U8_AA(x1, x2, x3)  =  U8_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)
U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_AAAAAAA(x8)
U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAAAA(x8)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA
U20_AAAA(x1, x2, x3, x4, x5, x6)  =  U20_AAAA(x6)
U12_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GAAAAAA(x1, x8)
U13_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GAAAAAA(x1, x8)
U14_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GAAAAAA(x1, x8)
U15_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GAAAAAA(x1, x8)
U16_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GAAAAAA(x1, x8)
U17_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_GAAAAAA(x1, x8)
U18_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_GAAAAAA(x1, x8)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)
U20_AGAA(x1, x2, x3, x4, x5, x6)  =  U20_AGAA(x3, x6)
U49_AG(x1, x2, x3, x4, x5, x6)  =  U49_AG(x1, x4, x6)
U50_AG(x1, x2, x3, x4, x5)  =  U50_AG(x4, x5)
P52_IN_AAAGG(x1, x2, x3, x4, x5)  =  P52_IN_AAAGG(x4, x5)
U33_AAAGG(x1, x2, x3, x4, x5, x6)  =  U33_AAAGG(x4, x5, x6)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
PART87_IN_AAAA(x1, x2, x3, x4)  =  PART87_IN_AAAA
U38_AAAA(x1, x2, x3, x4, x5, x6)  =  U38_AAAA(x6)
U39_AAAA(x1, x2, x3, x4, x5, x6)  =  U39_AAAA(x6)
U40_AAAA(x1, x2, x3, x4, x5, x6)  =  U40_AAAA(x6)
U41_AAAA(x1, x2, x3, x4, x5, x6)  =  U41_AAAA(x6)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U42_AAAA(x1, x2, x3, x4, x5, x6)  =  U42_AAAA(x6)
U43_AAAA(x1, x2, x3, x4, x5, x6)  =  U43_AAAA(x6)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA
U31_AAAA(x1, x2, x3, x4, x5, x6)  =  U31_AAAA(x6)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U34_AAAGG(x1, x2, x3, x4, x5, x6)  =  U34_AAAGG(x4, x5, x6)
U35_AAAGG(x1, x2, x3, x4, x5, x6)  =  U35_AAAGG(x4, x5, x6)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)
U32_AGAG(x1, x2, x3, x4, x5, x6)  =  U32_AGAG(x1, x3, x5, x6)
U51_AG(x1, x2, x3, x4, x5)  =  U51_AG(x4, x5)
U52_AG(x1, x2, x3, x4, x5)  =  U52_AG(x4, x5)
U53_AG(x1, x2, x3, x4, x5)  =  U53_AG(x4, x5)
U54_AG(x1, x2, x3, x4, x5)  =  U54_AG(x4, x5)
U55_AG(x1, x2, x3, x4, x5)  =  U55_AG(x1, x4, x5)
U56_AG(x1, x2, x3, x4, x5)  =  U56_AG(x4, x5)
U57_AG(x1, x2, x3, x4, x5, x6)  =  U57_AG(x1, x4, x6)
U58_AG(x1, x2, x3, x4, x5)  =  U58_AG(x4, x5)
U59_AG(x1, x2, x3)  =  U59_AG(x2, x3)
QS155_IN_A(x1)  =  QS155_IN_A
U60_AG(x1, x2, x3)  =  U60_AG(x2, x3)
U61_AG(x1, x2, x3)  =  U61_AG(x2, x3)
U62_AG(x1, x2, x3, x4)  =  U62_AG(x2, x3, x4)
U63_AG(x1, x2, x3)  =  U63_AG(x2, x3)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)
U36_GAGG(x1, x2, x3, x4, x5, x6)  =  U36_GAGG(x1, x2, x4, x5, 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_AG(.(T26, .(T27, T28)), T8) → U44_AG(T26, T27, T28, T8, gt10_in_aa(T26, T27))
QS1_IN_AG(.(T26, .(T27, T28)), T8) → GT10_IN_AA(T26, T27)
GT10_IN_AA(s(T46), s(T47)) → U1_AA(T46, T47, gt10_in_aa(T46, T47))
GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)
QS1_IN_AG(.(T31, .(T33, T32)), T8) → U45_AG(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → PART20_IN_GAAA(T31, T32, X47, X48)
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GAAA(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GA(T70, T73)
GT10_IN_GA(s(T46), s(T47)) → U1_GA(T46, T47, gt10_in_ga(T46, T47))
GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)
PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GAAA(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GA(T95, T98)
LE33_IN_GA(s(T113), s(T115)) → U8_GA(T113, T115, le33_in_ga(T113, T115))
LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_GAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_AG(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → QS51_IN_GAA(T33, T53, X9)
QS51_IN_GAA(T148, T149, X273) → U37_GAA(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
QS51_IN_GAA(T148, T149, X273) → P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273)
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_GAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_GAAA(T148, T149, X269, X270)
P56_IN_GAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → U19_AA(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_AAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_AAAA(T148, T149, X269, X270)
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_AAAA(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_AA(T70, T73)
PART20_IN_AAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_AAAA(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_AAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_AAAA(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_AA(T95, T98)
LE33_IN_AA(s(T113), s(T115)) → U8_AA(T113, T115, le33_in_aa(T113, T115))
LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)
PART20_IN_AAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_AAAA(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_AAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AAAA(T160, T148, T178, X273)
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AAAA(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AGAA(T160, T148, T178, X273)
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AGAA(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_AG(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_AG(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → P52_IN_AAAGG(T54, X10, T131, T31, T8)
P52_IN_AAAGG(T54, X10, T131, T31, T8) → U33_AAAGG(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
P52_IN_AAAGG(T54, X10, T131, T31, T8) → QS80_IN_AA(T54, X10)
QS80_IN_AA(.(T230, T231), X452) → U22_AA(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
QS80_IN_AA(.(T230, T231), X452) → PART87_IN_AAAA(T230, T231, X448, X449)
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → U38_AAAA(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → GT10_IN_AA(T256, T257)
PART87_IN_AAAA(T261, .(T257, T262), .(T257, X503), X504) → U39_AAAA(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_AAAA(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → PART20_IN_GAAA(T261, T262, X503, X504)
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → U41_AAAA(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → LE100_IN_AA(T283, T284)
LE100_IN_AA(s(T302), s(T303)) → U21_AA(T302, T303, le100_in_aa(T302, T303))
LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)
PART87_IN_AAAA(T288, .(T284, T289), X554, .(T284, X555)) → U42_AAAA(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_AAAA(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → PART20_IN_GAAA(T288, T289, X554, X555)
QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_AA(T237, T231, X452, qs80_in_aa(T235, X450))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_AA(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_AA(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → APP121_IN_AAAA(T319, T325, T324, X452)
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → U31_AAAA(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_AA(T237, T231, X452, qs80_in_aa(T236, X451))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)
P52_IN_AAAGG(T54, T217, T131, T31, T8) → U34_AAAGG(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_AAAGG(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → APP81_IN_AGAG(T131, T31, T217, T8)
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → U32_AGAG(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)
QS1_IN_AG(.(T421, .(T422, T423)), T8) → U51_AG(T421, T422, T423, T8, le100_in_aa(T421, T422))
QS1_IN_AG(.(T421, .(T422, T423)), T8) → LE100_IN_AA(T421, T422)
QS1_IN_AG(.(T426, .(T428, T427)), T8) → U52_AG(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_AG(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → PART20_IN_GAAA(T426, T427, X740, X741)
QS1_IN_AG(.(T426, .(T436, T427)), T8) → U54_AG(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_AG(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → PART20_IN_GAAA(T426, T427, T434, T435)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_AG(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → QS80_IN_AA(T434, X9)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_AG(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_AG(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → P52_IN_AAAGG(.(T436, T435), X10, T442, T426, T8)
QS1_IN_AG(.(T452, []), T8) → U59_AG(T452, T8, qs155_in_a(X9))
QS1_IN_AG(.(T452, []), T8) → QS155_IN_A(X9)
QS1_IN_AG(.(T452, []), T8) → U60_AG(T452, T8, qs155_in_a(T455))
U60_AG(T452, T8, qs155_out_a(T455)) → U61_AG(T452, T8, qs155_in_a(X10))
U60_AG(T452, T8, qs155_out_a(T455)) → QS155_IN_A(X10)
U60_AG(T452, T8, qs155_out_a(T455)) → U62_AG(T452, T8, T455, qs155_in_a(T460))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → U63_AG(T452, T8, app162_in_gagg(T455, T452, T460, T8))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → APP162_IN_GAGG(T455, T452, T460, T8)
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → U36_GAGG(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
QS1_IN_AG(x1, x2)  =  QS1_IN_AG(x2)
U44_AG(x1, x2, x3, x4, x5)  =  U44_AG(x4, x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U45_AG(x1, x2, x3, x4, x5)  =  U45_AG(x4, x5)
U46_AG(x1, x2, x3, x4, x5)  =  U46_AG(x4, x5)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
GT10_IN_GA(x1, x2)  =  GT10_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)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)
U47_AG(x1, x2, x3, x4, x5)  =  U47_AG(x1, x2, x4, x5)
U48_AG(x1, x2, x3, x4, x5)  =  U48_AG(x4, x5)
QS51_IN_GAA(x1, x2, x3)  =  QS51_IN_GAA(x1)
U37_GAA(x1, x2, x3, x4)  =  U37_GAA(x1, x4)
P56_IN_GAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_GAAAAAA(x1)
U9_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_GAAAAAA(x1, x8)
U10_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GAAAAAA(x1, x8)
U11_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GAAAAAA(x1, x8)
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U9_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AAAAAAA(x8)
PART20_IN_AAAA(x1, x2, x3, x4)  =  PART20_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)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U8_AA(x1, x2, x3)  =  U8_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)
U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_AAAAAAA(x8)
U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAAAA(x8)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA
U20_AAAA(x1, x2, x3, x4, x5, x6)  =  U20_AAAA(x6)
U12_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GAAAAAA(x1, x8)
U13_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GAAAAAA(x1, x8)
U14_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GAAAAAA(x1, x8)
U15_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GAAAAAA(x1, x8)
U16_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GAAAAAA(x1, x8)
U17_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_GAAAAAA(x1, x8)
U18_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_GAAAAAA(x1, x8)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)
U20_AGAA(x1, x2, x3, x4, x5, x6)  =  U20_AGAA(x3, x6)
U49_AG(x1, x2, x3, x4, x5, x6)  =  U49_AG(x1, x4, x6)
U50_AG(x1, x2, x3, x4, x5)  =  U50_AG(x4, x5)
P52_IN_AAAGG(x1, x2, x3, x4, x5)  =  P52_IN_AAAGG(x4, x5)
U33_AAAGG(x1, x2, x3, x4, x5, x6)  =  U33_AAAGG(x4, x5, x6)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
PART87_IN_AAAA(x1, x2, x3, x4)  =  PART87_IN_AAAA
U38_AAAA(x1, x2, x3, x4, x5, x6)  =  U38_AAAA(x6)
U39_AAAA(x1, x2, x3, x4, x5, x6)  =  U39_AAAA(x6)
U40_AAAA(x1, x2, x3, x4, x5, x6)  =  U40_AAAA(x6)
U41_AAAA(x1, x2, x3, x4, x5, x6)  =  U41_AAAA(x6)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U42_AAAA(x1, x2, x3, x4, x5, x6)  =  U42_AAAA(x6)
U43_AAAA(x1, x2, x3, x4, x5, x6)  =  U43_AAAA(x6)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA
U31_AAAA(x1, x2, x3, x4, x5, x6)  =  U31_AAAA(x6)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U34_AAAGG(x1, x2, x3, x4, x5, x6)  =  U34_AAAGG(x4, x5, x6)
U35_AAAGG(x1, x2, x3, x4, x5, x6)  =  U35_AAAGG(x4, x5, x6)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)
U32_AGAG(x1, x2, x3, x4, x5, x6)  =  U32_AGAG(x1, x3, x5, x6)
U51_AG(x1, x2, x3, x4, x5)  =  U51_AG(x4, x5)
U52_AG(x1, x2, x3, x4, x5)  =  U52_AG(x4, x5)
U53_AG(x1, x2, x3, x4, x5)  =  U53_AG(x4, x5)
U54_AG(x1, x2, x3, x4, x5)  =  U54_AG(x4, x5)
U55_AG(x1, x2, x3, x4, x5)  =  U55_AG(x1, x4, x5)
U56_AG(x1, x2, x3, x4, x5)  =  U56_AG(x4, x5)
U57_AG(x1, x2, x3, x4, x5, x6)  =  U57_AG(x1, x4, x6)
U58_AG(x1, x2, x3, x4, x5)  =  U58_AG(x4, x5)
U59_AG(x1, x2, x3)  =  U59_AG(x2, x3)
QS155_IN_A(x1)  =  QS155_IN_A
U60_AG(x1, x2, x3)  =  U60_AG(x2, x3)
U61_AG(x1, x2, x3)  =  U61_AG(x2, x3)
U62_AG(x1, x2, x3, x4)  =  U62_AG(x2, x3, x4)
U63_AG(x1, x2, x3)  =  U63_AG(x2, x3)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)
U36_GAGG(x1, x2, x3, x4, x5, x6)  =  U36_GAGG(x1, x2, x4, x5, 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 107 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)

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:

APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)

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:

APP162_IN_GAGG(.(T486, T487), T489, .(T486, T490)) → APP162_IN_GAGG(T487, T489, T490)

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:

• APP162_IN_GAGG(.(T486, T487), T489, .(T486, T490)) → APP162_IN_GAGG(T487, T489, T490)
  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:

APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)

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:

APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)

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:

APP81_IN_AGAG(T391, .(T389, T393)) → APP81_IN_AGAG(T391, T393)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP81_IN_AGAG(T391, .(T389, T393)) → APP81_IN_AGAG(T391, T393)
  The graph contains the following edges 1 >= 1, 2 > 2

(22) YES

(23) Obligation:

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

APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APP121_IN_AAAA → APP121_IN_AAAA

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

(28) NonTerminationProof (EQUIVALENT transformation)

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

s = APP121_IN_AAAA evaluates to t =APP121_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 APP121_IN_AAAA to APP121_IN_AAAA.

(29) NO

(30) Obligation:

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

LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA

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:

LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA

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:

LE100_IN_AA → LE100_IN_AA

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 = LE100_IN_AA evaluates to t =LE100_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 LE100_IN_AA to LE100_IN_AA.

(36) NO

(37) Obligation:

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

APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

APP70_IN_AGAA(T203) → APP70_IN_AGAA(T203)

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 = APP70_IN_AGAA(T203) evaluates to t =APP70_IN_AGAA(T203)

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 APP70_IN_AGAA(T203) to APP70_IN_AGAA(T203).

(43) NO

(44) Obligation:

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

APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA

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:

APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA

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:

APP70_IN_AAAA → APP70_IN_AAAA

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

(49) 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 = APP70_IN_AAAA evaluates to t =APP70_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 APP70_IN_AAAA to APP70_IN_AAAA.

(50) NO

(51) Obligation:

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

LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA

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:

LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA

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:

LE33_IN_AA → LE33_IN_AA

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

(56) 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 = LE33_IN_AA evaluates to t =LE33_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE33_IN_AA to LE33_IN_AA.

(57) NO

(58) Obligation:

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

LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)

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

(59) UsableRulesProof (EQUIVALENT transformation)

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

(60) Obligation:

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

LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)

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

(61) PiDPToQDPProof (SOUND transformation)

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

(62) Obligation:

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

LE33_IN_GA(s(T113)) → LE33_IN_GA(T113)

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

(63) 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:

• LE33_IN_GA(s(T113)) → LE33_IN_GA(T113)
  The graph contains the following edges 1 > 1

(64) YES

(65) Obligation:

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

GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

GT10_IN_GA(s(T46)) → GT10_IN_GA(T46)

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

(70) 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:

• GT10_IN_GA(s(T46)) → GT10_IN_GA(T46)
  The graph contains the following edges 1 > 1

(71) YES

(72) Obligation:

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

PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))

The argument filtering Pi contains the following mapping:
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

PART20_IN_GAAA(T70) → U3_GAAA(T70, gt10_in_ga(T70))
U3_GAAA(T70, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70)
PART20_IN_GAAA(T95) → U6_GAAA(T95, le33_in_ga(T95))
U6_GAAA(T95, le33_out_ga(T95)) → PART20_IN_GAAA(T95)

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(77) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T70) → U3_GAAA(T70, gt10_in_ga(T70)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))

(78) Obligation:

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

U3_GAAA(T70, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70)
PART20_IN_GAAA(T95) → U6_GAAA(T95, le33_in_ga(T95))
U6_GAAA(T95, le33_out_ga(T95)) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(79) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T95) → U6_GAAA(T95, le33_in_ga(T95)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))

(80) Obligation:

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

U3_GAAA(T70, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70)
U6_GAAA(T95, le33_out_ga(T95)) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(81) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAAA(T70, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70) we obtained the following new rules [LPAR04]:

U3_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))

(82) Obligation:

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

U6_GAAA(T95, le33_out_ga(T95)) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))
U3_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(83) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GAAA(T95, le33_out_ga(T95)) → PART20_IN_GAAA(T95) we obtained the following new rules [LPAR04]:

U6_GAAA(s(z0), le33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))
U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)

(84) Obligation:

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

PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))
U3_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
U6_GAAA(s(z0), le33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))
U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(85) DependencyGraphProof (EQUIVALENT transformation)

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

(86) Complex Obligation (AND)

(87) Obligation:

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

U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(88) UsableRulesProof (EQUIVALENT transformation)

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

(89) Obligation:

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

U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))

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

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(90) QReductionProof (EQUIVALENT transformation)

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

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

(91) Obligation:

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

U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))

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

(92) 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 = PART20_IN_GAAA(0) evaluates to t =PART20_IN_GAAA(0)

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0))
with rule PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga(0)) at position [] and matcher [ ]

U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
with rule U6_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)

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


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

(93) NO

(94) Obligation:

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

U3_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
U3_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(x0, le33_in_ga(x0)))
U6_GAAA(s(z0), le33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)

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

(95) 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 = PART20_IN_GAAA(s(0)) evaluates to t =PART20_IN_GAAA(s(0))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0))
with rule PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(s(0), 0)) at position [] and matcher [ ]

U3_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
with rule U3_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))

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


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

(96) NO

(97) Obligation:

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

GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA

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

(98) UsableRulesProof (EQUIVALENT transformation)

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

(99) Obligation:

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

GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA

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

(100) PiDPToQDPProof (SOUND transformation)

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

(101) Obligation:

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

GT10_IN_AA → GT10_IN_AA

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

(102) 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 = GT10_IN_AA evaluates to t =GT10_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT10_IN_AA to GT10_IN_AA.

(103) NO

(104) Obligation:

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

QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)

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

(105) UsableRulesProof (EQUIVALENT transformation)

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

(106) Obligation:

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

QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)

The TRS R consists of the following rules:

part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
qs80_in_aa([], []) → qs80_out_aa([], [])
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))

The argument filtering Pi contains the following mapping:
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
.(x1, x2)  =  .(x1, x2)
[]  =  []
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)

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

(107) PiDPToQDPProof (SOUND transformation)

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

(108) Obligation:

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

QS80_IN_AA → U23_AA(part87_in_aaaa)
U23_AA(part87_out_aaaa) → QS80_IN_AA
QS80_IN_AA → U27_AA(part87_in_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa)
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa(T261)) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa(T288)) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
app121_in_aaaa
U1_ga(x0, x1)
U8_ga(x0, x1)
U31_aaaa(x0)

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

(109) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS80_IN_AA → U23_AA(part87_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)

(110) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
QS80_IN_AA → U27_AA(part87_in_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa)
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa(T261)) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa(T288)) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
app121_in_aaaa
U1_ga(x0, x1)
U8_ga(x0, x1)
U31_aaaa(x0)

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

(111) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS80_IN_AA → U27_AA(part87_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)

(112) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa)
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)
QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa(T261)) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa(T288)) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
app121_in_aaaa
U1_ga(x0, x1)
U8_ga(x0, x1)
U31_aaaa(x0)

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

(113) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U27_AA(part87_out_aaaa) → U28_AA(U22_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U23_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U27_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(qs80_out_aa)

(114) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)
QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(U22_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U23_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U27_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(qs80_out_aa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa(T261)) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa(T288)) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
app121_in_aaaa
U1_ga(x0, x1)
U8_ga(x0, x1)
U31_aaaa(x0)

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

(115) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U23_AA(part87_out_aaaa) → U25_AA(U22_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U23_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U27_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(qs80_out_aa)

(116) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)
QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(U22_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U23_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U27_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(qs80_out_aa)
U23_AA(part87_out_aaaa) → U25_AA(U22_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U23_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U27_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(qs80_out_aa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa(T261)) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa(T288)) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
app121_in_aaaa
U1_ga(x0, x1)
U8_ga(x0, x1)
U31_aaaa(x0)

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

(117) 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 = QS80_IN_AA evaluates to t =QS80_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

QS80_IN_AA → U23_AA(part87_out_aaaa)
with rule QS80_IN_AA → U23_AA(part87_out_aaaa) at position [] and matcher [ ]

U23_AA(part87_out_aaaa) → QS80_IN_AA
with rule U23_AA(part87_out_aaaa) → QS80_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.

(118) NO

(119) Obligation:

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

P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x4, x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag(x2)
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x4, x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x4, x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x1, x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x1, x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa(x1)
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x1, x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x1, x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x1, x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x1, x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x1, x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x1, x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa(x2)
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa(x1)
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x4, x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x4, x5, x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg(x4, x5)
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x4, x5, x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x3, x5, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x2, x3, x4)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x4, x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x4, x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x4, x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x4, x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x4, x5)
U59_ag(x1, x2, x3)  =  U59_ag(x2, x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x2, x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x2, x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x1, x2, x4, x5, x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x1, x2, x3, x4)
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)

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

(120) UsableRulesProof (EQUIVALENT transformation)

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

(121) Obligation:

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

P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)

The TRS R consists of the following rules:

part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
qs59_in_aa([], []) → qs59_out_aa([], [])
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))

The argument filtering Pi contains the following mapping:
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x1, x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_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)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)

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

(122) PiDPToQDPProof (SOUND transformation)

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

(123) Obligation:

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

P56_IN_AAAAAAA → U10_AAAAAAA(part20_in_aaaa)
U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
P56_IN_AAAAAAA → U12_AAAAAAA(part20_in_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa)
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa(T70)) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa(T95)) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(124) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P56_IN_AAAAAAA → U10_AAAAAAA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)

(125) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
P56_IN_AAAAAAA → U12_AAAAAAA(part20_in_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa)
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa(T70)) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa(T95)) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(126) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P56_IN_AAAAAAA → U12_AAAAAAA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)

(127) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa)
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa(T70)) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa(T95)) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(128) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)

(129) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa(T70)) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa(T95)) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(130) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P56_IN_AAAAAAA → U15_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(part20_out_aaaa)

(131) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)
P56_IN_AAAAAAA → U15_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa(T70)) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa(T95)) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(132) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_out_aa)

(133) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)
P56_IN_AAAAAAA → U15_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(part20_out_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_out_aa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa(T70)) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa(T95)) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa(T128)
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(T70, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70)
U3_gaaa(T70, gt10_out_ga(T70, T73)) → U4_gaaa(T70, part20_in_gaaa(T70))
U5_gaaa(T95, le33_out_ga(T95)) → part20_out_gaaa(T95)
U6_gaaa(T95, le33_out_ga(T95)) → U7_gaaa(T95, part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(T46, gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, part20_out_gaaa(T70)) → part20_out_gaaa(T70)
le33_in_ga(s(T113)) → U8_ga(T113, le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga(0)
U7_gaaa(T95, part20_out_gaaa(T95)) → part20_out_gaaa(T95)
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(T46, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, le33_out_ga(T113)) → le33_out_ga(s(T113))
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
U5_gaaa(x0, x1)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0, x1)
le33_in_ga(x0)
U7_gaaa(x0, x1)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0, x1)
U8_ga(x0, x1)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(134) 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 = QS59_IN_AA evaluates to t =QS59_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS59_IN_AA → P56_IN_AAAAAAA
with rule QS59_IN_AA → P56_IN_AAAAAAA at position [] and matcher [ ]

P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
with rule P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa) at position [] and matcher [ ]

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
with rule U10_AAAAAAA(part20_out_aaaa) → QS59_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.

(135) NO

(136) 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: (f,b)
gt10_in: (f,f) (b,f)
part20_in: (b,f,f,f) (f,f,f,f)
le33_in: (b,f) (f,f)
qs51_in: (b,f,f)
p56_in: (b,f,f,f,f,f,f) (f,f,f,f,f,f,f)
qs59_in: (f,f)
app70_in: (f,f,f,f) (f,b,f,f)
p52_in: (f,f,f,b,b)
qs80_in: (f,f)
part87_in: (f,f,f,f)
le100_in: (f,f)
app121_in: (f,f,f,f)
app81_in: (f,b,f,b)
app162_in: (b,f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(137) Obligation:

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

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)

(138) 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_AG(.(T26, .(T27, T28)), T8) → U44_AG(T26, T27, T28, T8, gt10_in_aa(T26, T27))
QS1_IN_AG(.(T26, .(T27, T28)), T8) → GT10_IN_AA(T26, T27)
GT10_IN_AA(s(T46), s(T47)) → U1_AA(T46, T47, gt10_in_aa(T46, T47))
GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)
QS1_IN_AG(.(T31, .(T33, T32)), T8) → U45_AG(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → PART20_IN_GAAA(T31, T32, X47, X48)
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GAAA(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GA(T70, T73)
GT10_IN_GA(s(T46), s(T47)) → U1_GA(T46, T47, gt10_in_ga(T46, T47))
GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)
PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GAAA(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GA(T95, T98)
LE33_IN_GA(s(T113), s(T115)) → U8_GA(T113, T115, le33_in_ga(T113, T115))
LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_GAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_AG(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → QS51_IN_GAA(T33, T53, X9)
QS51_IN_GAA(T148, T149, X273) → U37_GAA(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
QS51_IN_GAA(T148, T149, X273) → P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273)
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_GAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_GAAA(T148, T149, X269, X270)
P56_IN_GAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → U19_AA(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_AAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_AAAA(T148, T149, X269, X270)
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_AAAA(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_AA(T70, T73)
PART20_IN_AAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_AAAA(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_AAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_AAAA(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_AA(T95, T98)
LE33_IN_AA(s(T113), s(T115)) → U8_AA(T113, T115, le33_in_aa(T113, T115))
LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)
PART20_IN_AAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_AAAA(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_AAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AAAA(T160, T148, T178, X273)
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AAAA(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AGAA(T160, T148, T178, X273)
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AGAA(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_AG(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_AG(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → P52_IN_AAAGG(T54, X10, T131, T31, T8)
P52_IN_AAAGG(T54, X10, T131, T31, T8) → U33_AAAGG(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
P52_IN_AAAGG(T54, X10, T131, T31, T8) → QS80_IN_AA(T54, X10)
QS80_IN_AA(.(T230, T231), X452) → U22_AA(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
QS80_IN_AA(.(T230, T231), X452) → PART87_IN_AAAA(T230, T231, X448, X449)
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → U38_AAAA(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → GT10_IN_AA(T256, T257)
PART87_IN_AAAA(T261, .(T257, T262), .(T257, X503), X504) → U39_AAAA(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_AAAA(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → PART20_IN_GAAA(T261, T262, X503, X504)
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → U41_AAAA(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → LE100_IN_AA(T283, T284)
LE100_IN_AA(s(T302), s(T303)) → U21_AA(T302, T303, le100_in_aa(T302, T303))
LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)
PART87_IN_AAAA(T288, .(T284, T289), X554, .(T284, X555)) → U42_AAAA(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_AAAA(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → PART20_IN_GAAA(T288, T289, X554, X555)
QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_AA(T237, T231, X452, qs80_in_aa(T235, X450))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_AA(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_AA(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → APP121_IN_AAAA(T319, T325, T324, X452)
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → U31_AAAA(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_AA(T237, T231, X452, qs80_in_aa(T236, X451))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)
P52_IN_AAAGG(T54, T217, T131, T31, T8) → U34_AAAGG(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_AAAGG(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → APP81_IN_AGAG(T131, T31, T217, T8)
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → U32_AGAG(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)
QS1_IN_AG(.(T421, .(T422, T423)), T8) → U51_AG(T421, T422, T423, T8, le100_in_aa(T421, T422))
QS1_IN_AG(.(T421, .(T422, T423)), T8) → LE100_IN_AA(T421, T422)
QS1_IN_AG(.(T426, .(T428, T427)), T8) → U52_AG(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_AG(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → PART20_IN_GAAA(T426, T427, X740, X741)
QS1_IN_AG(.(T426, .(T436, T427)), T8) → U54_AG(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_AG(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → PART20_IN_GAAA(T426, T427, T434, T435)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_AG(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → QS80_IN_AA(T434, X9)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_AG(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_AG(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → P52_IN_AAAGG(.(T436, T435), X10, T442, T426, T8)
QS1_IN_AG(.(T452, []), T8) → U59_AG(T452, T8, qs155_in_a(X9))
QS1_IN_AG(.(T452, []), T8) → QS155_IN_A(X9)
QS1_IN_AG(.(T452, []), T8) → U60_AG(T452, T8, qs155_in_a(T455))
U60_AG(T452, T8, qs155_out_a(T455)) → U61_AG(T452, T8, qs155_in_a(X10))
U60_AG(T452, T8, qs155_out_a(T455)) → QS155_IN_A(X10)
U60_AG(T452, T8, qs155_out_a(T455)) → U62_AG(T452, T8, T455, qs155_in_a(T460))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → U63_AG(T452, T8, app162_in_gagg(T455, T452, T460, T8))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → APP162_IN_GAGG(T455, T452, T460, T8)
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → U36_GAGG(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
QS1_IN_AG(x1, x2)  =  QS1_IN_AG(x2)
U44_AG(x1, x2, x3, x4, x5)  =  U44_AG(x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U45_AG(x1, x2, x3, x4, x5)  =  U45_AG(x4, x5)
U46_AG(x1, x2, x3, x4, x5)  =  U46_AG(x5)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x6)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x6)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x6)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x6)
U47_AG(x1, x2, x3, x4, x5)  =  U47_AG(x1, x2, x4, x5)
U48_AG(x1, x2, x3, x4, x5)  =  U48_AG(x5)
QS51_IN_GAA(x1, x2, x3)  =  QS51_IN_GAA(x1)
U37_GAA(x1, x2, x3, x4)  =  U37_GAA(x4)
P56_IN_GAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_GAAAAAA(x1)
U9_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_GAAAAAA(x8)
U10_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GAAAAAA(x8)
U11_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GAAAAAA(x8)
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U9_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AAAAAAA(x8)
PART20_IN_AAAA(x1, x2, x3, x4)  =  PART20_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)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U8_AA(x1, x2, x3)  =  U8_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)
U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_AAAAAAA(x8)
U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAAAA(x8)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA
U20_AAAA(x1, x2, x3, x4, x5, x6)  =  U20_AAAA(x6)
U12_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GAAAAAA(x8)
U13_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GAAAAAA(x8)
U14_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GAAAAAA(x8)
U15_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GAAAAAA(x1, x8)
U16_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GAAAAAA(x1, x8)
U17_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_GAAAAAA(x1, x8)
U18_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_GAAAAAA(x8)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)
U20_AGAA(x1, x2, x3, x4, x5, x6)  =  U20_AGAA(x6)
U49_AG(x1, x2, x3, x4, x5, x6)  =  U49_AG(x1, x4, x6)
U50_AG(x1, x2, x3, x4, x5)  =  U50_AG(x5)
P52_IN_AAAGG(x1, x2, x3, x4, x5)  =  P52_IN_AAAGG(x4, x5)
U33_AAAGG(x1, x2, x3, x4, x5, x6)  =  U33_AAAGG(x6)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
PART87_IN_AAAA(x1, x2, x3, x4)  =  PART87_IN_AAAA
U38_AAAA(x1, x2, x3, x4, x5, x6)  =  U38_AAAA(x6)
U39_AAAA(x1, x2, x3, x4, x5, x6)  =  U39_AAAA(x6)
U40_AAAA(x1, x2, x3, x4, x5, x6)  =  U40_AAAA(x6)
U41_AAAA(x1, x2, x3, x4, x5, x6)  =  U41_AAAA(x6)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U42_AAAA(x1, x2, x3, x4, x5, x6)  =  U42_AAAA(x6)
U43_AAAA(x1, x2, x3, x4, x5, x6)  =  U43_AAAA(x6)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA
U31_AAAA(x1, x2, x3, x4, x5, x6)  =  U31_AAAA(x6)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U34_AAAGG(x1, x2, x3, x4, x5, x6)  =  U34_AAAGG(x4, x5, x6)
U35_AAAGG(x1, x2, x3, x4, x5, x6)  =  U35_AAAGG(x6)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)
U32_AGAG(x1, x2, x3, x4, x5, x6)  =  U32_AGAG(x1, x6)
U51_AG(x1, x2, x3, x4, x5)  =  U51_AG(x5)
U52_AG(x1, x2, x3, x4, x5)  =  U52_AG(x5)
U53_AG(x1, x2, x3, x4, x5)  =  U53_AG(x5)
U54_AG(x1, x2, x3, x4, x5)  =  U54_AG(x4, x5)
U55_AG(x1, x2, x3, x4, x5)  =  U55_AG(x1, x4, x5)
U56_AG(x1, x2, x3, x4, x5)  =  U56_AG(x5)
U57_AG(x1, x2, x3, x4, x5, x6)  =  U57_AG(x1, x4, x6)
U58_AG(x1, x2, x3, x4, x5)  =  U58_AG(x5)
U59_AG(x1, x2, x3)  =  U59_AG(x3)
QS155_IN_A(x1)  =  QS155_IN_A
U60_AG(x1, x2, x3)  =  U60_AG(x2, x3)
U61_AG(x1, x2, x3)  =  U61_AG(x3)
U62_AG(x1, x2, x3, x4)  =  U62_AG(x2, x3, x4)
U63_AG(x1, x2, x3)  =  U63_AG(x3)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)
U36_GAGG(x1, x2, x3, x4, x5, x6)  =  U36_GAGG(x6)

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

(139) Obligation:

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

QS1_IN_AG(.(T26, .(T27, T28)), T8) → U44_AG(T26, T27, T28, T8, gt10_in_aa(T26, T27))
QS1_IN_AG(.(T26, .(T27, T28)), T8) → GT10_IN_AA(T26, T27)
GT10_IN_AA(s(T46), s(T47)) → U1_AA(T46, T47, gt10_in_aa(T46, T47))
GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)
QS1_IN_AG(.(T31, .(T33, T32)), T8) → U45_AG(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → PART20_IN_GAAA(T31, T32, X47, X48)
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GAAA(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GA(T70, T73)
GT10_IN_GA(s(T46), s(T47)) → U1_GA(T46, T47, gt10_in_ga(T46, T47))
GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)
PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GAAA(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GA(T95, T98)
LE33_IN_GA(s(T113), s(T115)) → U8_GA(T113, T115, le33_in_ga(T113, T115))
LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_GAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
U45_AG(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_AG(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → QS51_IN_GAA(T33, T53, X9)
QS51_IN_GAA(T148, T149, X273) → U37_GAA(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
QS51_IN_GAA(T148, T149, X273) → P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273)
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_GAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
P56_IN_GAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_GAAA(T148, T149, X269, X270)
P56_IN_GAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_GAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → U19_AA(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_AAAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
P56_IN_AAAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_AAAA(T148, T149, X269, X270)
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_AAAA(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
PART20_IN_AAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_AA(T70, T73)
PART20_IN_AAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_AAAA(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_AAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_AAAA(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_AAAA(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
PART20_IN_AAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_AA(T95, T98)
LE33_IN_AA(s(T113), s(T115)) → U8_AA(T113, T115, le33_in_aa(T113, T115))
LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)
PART20_IN_AAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_AAAA(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_AAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_AAAA(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
U17_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AAAA(T160, T148, T178, X273)
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AAAA(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U12_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U13_GAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
P56_IN_GAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U15_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U16_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
U17_GAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → APP70_IN_AGAA(T160, T148, T178, X273)
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → U20_AGAA(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)
U47_AG(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_AG(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_AG(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
U49_AG(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → P52_IN_AAAGG(T54, X10, T131, T31, T8)
P52_IN_AAAGG(T54, X10, T131, T31, T8) → U33_AAAGG(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
P52_IN_AAAGG(T54, X10, T131, T31, T8) → QS80_IN_AA(T54, X10)
QS80_IN_AA(.(T230, T231), X452) → U22_AA(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
QS80_IN_AA(.(T230, T231), X452) → PART87_IN_AAAA(T230, T231, X448, X449)
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → U38_AAAA(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
PART87_IN_AAAA(T256, .(T257, T258), .(T257, X503), X504) → GT10_IN_AA(T256, T257)
PART87_IN_AAAA(T261, .(T257, T262), .(T257, X503), X504) → U39_AAAA(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_AAAA(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U39_AAAA(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → PART20_IN_GAAA(T261, T262, X503, X504)
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → U41_AAAA(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
PART87_IN_AAAA(T283, .(T284, T285), X554, .(T284, X555)) → LE100_IN_AA(T283, T284)
LE100_IN_AA(s(T302), s(T303)) → U21_AA(T302, T303, le100_in_aa(T302, T303))
LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)
PART87_IN_AAAA(T288, .(T284, T289), X554, .(T284, X555)) → U42_AAAA(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_AAAA(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U42_AAAA(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → PART20_IN_GAAA(T288, T289, X554, X555)
QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_AA(T237, T231, X452, qs80_in_aa(T235, X450))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_AA(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_AA(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
U29_AA(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → APP121_IN_AAAA(T319, T325, T324, X452)
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → U31_AAAA(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_AA(T237, T231, X452, qs80_in_aa(T236, X451))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)
P52_IN_AAAGG(T54, T217, T131, T31, T8) → U34_AAAGG(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_AAAGG(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
U34_AAAGG(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → APP81_IN_AGAG(T131, T31, T217, T8)
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → U32_AGAG(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)
QS1_IN_AG(.(T421, .(T422, T423)), T8) → U51_AG(T421, T422, T423, T8, le100_in_aa(T421, T422))
QS1_IN_AG(.(T421, .(T422, T423)), T8) → LE100_IN_AA(T421, T422)
QS1_IN_AG(.(T426, .(T428, T427)), T8) → U52_AG(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_AG(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U52_AG(T426, T428, T427, T8, le100_out_aa(T426, T428)) → PART20_IN_GAAA(T426, T427, X740, X741)
QS1_IN_AG(.(T426, .(T436, T427)), T8) → U54_AG(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_AG(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U54_AG(T426, T436, T427, T8, le100_out_aa(T426, T436)) → PART20_IN_GAAA(T426, T427, T434, T435)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_AG(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → QS80_IN_AA(T434, X9)
U55_AG(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_AG(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_AG(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U57_AG(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → P52_IN_AAAGG(.(T436, T435), X10, T442, T426, T8)
QS1_IN_AG(.(T452, []), T8) → U59_AG(T452, T8, qs155_in_a(X9))
QS1_IN_AG(.(T452, []), T8) → QS155_IN_A(X9)
QS1_IN_AG(.(T452, []), T8) → U60_AG(T452, T8, qs155_in_a(T455))
U60_AG(T452, T8, qs155_out_a(T455)) → U61_AG(T452, T8, qs155_in_a(X10))
U60_AG(T452, T8, qs155_out_a(T455)) → QS155_IN_A(X10)
U60_AG(T452, T8, qs155_out_a(T455)) → U62_AG(T452, T8, T455, qs155_in_a(T460))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → U63_AG(T452, T8, app162_in_gagg(T455, T452, T460, T8))
U62_AG(T452, T8, T455, qs155_out_a(T460)) → APP162_IN_GAGG(T455, T452, T460, T8)
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → U36_GAGG(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
QS1_IN_AG(x1, x2)  =  QS1_IN_AG(x2)
U44_AG(x1, x2, x3, x4, x5)  =  U44_AG(x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U45_AG(x1, x2, x3, x4, x5)  =  U45_AG(x4, x5)
U46_AG(x1, x2, x3, x4, x5)  =  U46_AG(x5)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x6)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x6)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x6)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x6)
U47_AG(x1, x2, x3, x4, x5)  =  U47_AG(x1, x2, x4, x5)
U48_AG(x1, x2, x3, x4, x5)  =  U48_AG(x5)
QS51_IN_GAA(x1, x2, x3)  =  QS51_IN_GAA(x1)
U37_GAA(x1, x2, x3, x4)  =  U37_GAA(x4)
P56_IN_GAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_GAAAAAA(x1)
U9_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_GAAAAAA(x8)
U10_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GAAAAAA(x8)
U11_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GAAAAAA(x8)
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U9_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AAAAAAA(x8)
PART20_IN_AAAA(x1, x2, x3, x4)  =  PART20_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)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U8_AA(x1, x2, x3)  =  U8_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)
U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_AAAAAAA(x8)
U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAAAA(x8)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA
U20_AAAA(x1, x2, x3, x4, x5, x6)  =  U20_AAAA(x6)
U12_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GAAAAAA(x8)
U13_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GAAAAAA(x8)
U14_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GAAAAAA(x8)
U15_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GAAAAAA(x1, x8)
U16_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GAAAAAA(x1, x8)
U17_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_GAAAAAA(x1, x8)
U18_GAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_GAAAAAA(x8)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)
U20_AGAA(x1, x2, x3, x4, x5, x6)  =  U20_AGAA(x6)
U49_AG(x1, x2, x3, x4, x5, x6)  =  U49_AG(x1, x4, x6)
U50_AG(x1, x2, x3, x4, x5)  =  U50_AG(x5)
P52_IN_AAAGG(x1, x2, x3, x4, x5)  =  P52_IN_AAAGG(x4, x5)
U33_AAAGG(x1, x2, x3, x4, x5, x6)  =  U33_AAAGG(x6)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
PART87_IN_AAAA(x1, x2, x3, x4)  =  PART87_IN_AAAA
U38_AAAA(x1, x2, x3, x4, x5, x6)  =  U38_AAAA(x6)
U39_AAAA(x1, x2, x3, x4, x5, x6)  =  U39_AAAA(x6)
U40_AAAA(x1, x2, x3, x4, x5, x6)  =  U40_AAAA(x6)
U41_AAAA(x1, x2, x3, x4, x5, x6)  =  U41_AAAA(x6)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U42_AAAA(x1, x2, x3, x4, x5, x6)  =  U42_AAAA(x6)
U43_AAAA(x1, x2, x3, x4, x5, x6)  =  U43_AAAA(x6)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA
U31_AAAA(x1, x2, x3, x4, x5, x6)  =  U31_AAAA(x6)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U34_AAAGG(x1, x2, x3, x4, x5, x6)  =  U34_AAAGG(x4, x5, x6)
U35_AAAGG(x1, x2, x3, x4, x5, x6)  =  U35_AAAGG(x6)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)
U32_AGAG(x1, x2, x3, x4, x5, x6)  =  U32_AGAG(x1, x6)
U51_AG(x1, x2, x3, x4, x5)  =  U51_AG(x5)
U52_AG(x1, x2, x3, x4, x5)  =  U52_AG(x5)
U53_AG(x1, x2, x3, x4, x5)  =  U53_AG(x5)
U54_AG(x1, x2, x3, x4, x5)  =  U54_AG(x4, x5)
U55_AG(x1, x2, x3, x4, x5)  =  U55_AG(x1, x4, x5)
U56_AG(x1, x2, x3, x4, x5)  =  U56_AG(x5)
U57_AG(x1, x2, x3, x4, x5, x6)  =  U57_AG(x1, x4, x6)
U58_AG(x1, x2, x3, x4, x5)  =  U58_AG(x5)
U59_AG(x1, x2, x3)  =  U59_AG(x3)
QS155_IN_A(x1)  =  QS155_IN_A
U60_AG(x1, x2, x3)  =  U60_AG(x2, x3)
U61_AG(x1, x2, x3)  =  U61_AG(x3)
U62_AG(x1, x2, x3, x4)  =  U62_AG(x2, x3, x4)
U63_AG(x1, x2, x3)  =  U63_AG(x3)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)
U36_GAGG(x1, x2, x3, x4, x5, x6)  =  U36_GAGG(x6)

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

(140) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 13 SCCs with 107 less nodes.

(141) Complex Obligation (AND)

(142) Obligation:

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

APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)

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

(143) UsableRulesProof (EQUIVALENT transformation)

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

(144) Obligation:

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

APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP162_IN_GAGG(x1, x2, x3, x4)  =  APP162_IN_GAGG(x1, x3, x4)

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

(145) PiDPToQDPProof (SOUND transformation)

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

(146) Obligation:

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

APP162_IN_GAGG(.(T486, T487), T489, .(T486, T490)) → APP162_IN_GAGG(T487, T489, T490)

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

(147) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP162_IN_GAGG(.(T486, T487), T489, .(T486, T490)) → APP162_IN_GAGG(T487, T489, T490)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(148) YES

(149) Obligation:

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

APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)

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

(150) UsableRulesProof (EQUIVALENT transformation)

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

(151) Obligation:

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

APP81_IN_AGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_AGAG(T390, T391, T394, T393)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP81_IN_AGAG(x1, x2, x3, x4)  =  APP81_IN_AGAG(x2, x4)

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

(152) PiDPToQDPProof (SOUND transformation)

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

(153) Obligation:

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

APP81_IN_AGAG(T391, .(T389, T393)) → APP81_IN_AGAG(T391, T393)

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

(154) 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:

• APP81_IN_AGAG(T391, .(T389, T393)) → APP81_IN_AGAG(T391, T393)
  The graph contains the following edges 1 >= 1, 2 > 2

(155) YES

(156) Obligation:

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

APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA

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

(157) UsableRulesProof (EQUIVALENT transformation)

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

(158) Obligation:

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

APP121_IN_AAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_AAAA(T349, T352, T353, X657)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP121_IN_AAAA(x1, x2, x3, x4)  =  APP121_IN_AAAA

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

(159) PiDPToQDPProof (SOUND transformation)

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

(160) Obligation:

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

APP121_IN_AAAA → APP121_IN_AAAA

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

(161) 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 = APP121_IN_AAAA evaluates to t =APP121_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 APP121_IN_AAAA to APP121_IN_AAAA.

(162) NO

(163) Obligation:

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

LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA

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

(164) UsableRulesProof (EQUIVALENT transformation)

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

(165) Obligation:

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

LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE100_IN_AA(x1, x2)  =  LE100_IN_AA

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

(166) PiDPToQDPProof (SOUND transformation)

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

(167) Obligation:

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

LE100_IN_AA → LE100_IN_AA

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

(168) 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 = LE100_IN_AA evaluates to t =LE100_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 LE100_IN_AA to LE100_IN_AA.

(169) NO

(170) Obligation:

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

APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)

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

(171) UsableRulesProof (EQUIVALENT transformation)

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

(172) Obligation:

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

APP70_IN_AGAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AGAA(T202, T203, T204, X391)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP70_IN_AGAA(x1, x2, x3, x4)  =  APP70_IN_AGAA(x2)

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

(173) PiDPToQDPProof (SOUND transformation)

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

(174) Obligation:

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

APP70_IN_AGAA(T203) → APP70_IN_AGAA(T203)

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

(175) 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 = APP70_IN_AGAA(T203) evaluates to t =APP70_IN_AGAA(T203)

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 APP70_IN_AGAA(T203) to APP70_IN_AGAA(T203).

(176) NO

(177) Obligation:

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

APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA

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

(178) UsableRulesProof (EQUIVALENT transformation)

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

(179) Obligation:

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

APP70_IN_AAAA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_AAAA(T202, T203, T204, X391)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP70_IN_AAAA(x1, x2, x3, x4)  =  APP70_IN_AAAA

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

(180) PiDPToQDPProof (SOUND transformation)

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

(181) Obligation:

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

APP70_IN_AAAA → APP70_IN_AAAA

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

(182) 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 = APP70_IN_AAAA evaluates to t =APP70_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 APP70_IN_AAAA to APP70_IN_AAAA.

(183) NO

(184) Obligation:

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

LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA

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

(185) UsableRulesProof (EQUIVALENT transformation)

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

(186) Obligation:

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

LE33_IN_AA(s(T113), s(T115)) → LE33_IN_AA(T113, T115)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA

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

(187) PiDPToQDPProof (SOUND transformation)

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

(188) Obligation:

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

LE33_IN_AA → LE33_IN_AA

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

(189) 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 = LE33_IN_AA evaluates to t =LE33_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 LE33_IN_AA to LE33_IN_AA.

(190) NO

(191) Obligation:

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

LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)

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

(192) UsableRulesProof (EQUIVALENT transformation)

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

(193) Obligation:

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

LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)

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

(194) PiDPToQDPProof (SOUND transformation)

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

(195) Obligation:

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

LE33_IN_GA(s(T113)) → LE33_IN_GA(T113)

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

(196) 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:

• LE33_IN_GA(s(T113)) → LE33_IN_GA(T113)
  The graph contains the following edges 1 > 1

(197) YES

(198) Obligation:

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

GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)

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

(199) UsableRulesProof (EQUIVALENT transformation)

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

(200) Obligation:

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

GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)

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

(201) PiDPToQDPProof (SOUND transformation)

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

(202) Obligation:

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

GT10_IN_GA(s(T46)) → GT10_IN_GA(T46)

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

(203) 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:

• GT10_IN_GA(s(T46)) → GT10_IN_GA(T46)
  The graph contains the following edges 1 > 1

(204) YES

(205) Obligation:

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

PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)

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

(206) UsableRulesProof (EQUIVALENT transformation)

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

(207) Obligation:

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

PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))

The argument filtering Pi contains the following mapping:
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
.(x1, x2)  =  .(x1, x2)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)

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

(208) PiDPToQDPProof (SOUND transformation)

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

(209) Obligation:

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

PART20_IN_GAAA(T70) → U3_GAAA(T70, gt10_in_ga(T70))
U3_GAAA(T70, gt10_out_ga(T73)) → PART20_IN_GAAA(T70)
PART20_IN_GAAA(T95) → U6_GAAA(T95, le33_in_ga(T95))
U6_GAAA(T95, le33_out_ga) → PART20_IN_GAAA(T95)

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(210) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T70) → U3_GAAA(T70, gt10_in_ga(T70)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))

(211) Obligation:

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

U3_GAAA(T70, gt10_out_ga(T73)) → PART20_IN_GAAA(T70)
PART20_IN_GAAA(T95) → U6_GAAA(T95, le33_in_ga(T95))
U6_GAAA(T95, le33_out_ga) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(212) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T95) → U6_GAAA(T95, le33_in_ga(T95)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)

(213) Obligation:

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

U3_GAAA(T70, gt10_out_ga(T73)) → PART20_IN_GAAA(T70)
U6_GAAA(T95, le33_out_ga) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(214) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAAA(T70, gt10_out_ga(T73)) → PART20_IN_GAAA(T70) we obtained the following new rules [LPAR04]:

U3_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))

(215) Obligation:

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

U6_GAAA(T95, le33_out_ga) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)
U3_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(216) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GAAA(T95, le33_out_ga) → PART20_IN_GAAA(T95) we obtained the following new rules [LPAR04]:

U6_GAAA(s(z0), le33_out_ga) → PART20_IN_GAAA(s(z0))
U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)

(217) Obligation:

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

PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)
U3_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))
U6_GAAA(s(z0), le33_out_ga) → PART20_IN_GAAA(s(z0))
U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(218) DependencyGraphProof (EQUIVALENT transformation)

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

(219) Complex Obligation (AND)

(220) Obligation:

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

U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(221) UsableRulesProof (EQUIVALENT transformation)

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

(222) Obligation:

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

U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)

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

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(223) QReductionProof (EQUIVALENT transformation)

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

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

(224) Obligation:

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

U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)

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

(225) 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 = PART20_IN_GAAA(0) evaluates to t =PART20_IN_GAAA(0)

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga)
with rule PART20_IN_GAAA(0) → U6_GAAA(0, le33_out_ga) at position [] and matcher [ ]

U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
with rule U6_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)

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


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

(226) NO

(227) Obligation:

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

U3_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
U3_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U8_ga(le33_in_ga(x0)))
U6_GAAA(s(z0), le33_out_ga) → PART20_IN_GAAA(s(z0))

The TRS R consists of the following rules:

gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0)
U8_ga(x0)

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

(228) 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 = PART20_IN_GAAA(s(0)) evaluates to t =PART20_IN_GAAA(s(0))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0))
with rule PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gt10_out_ga(0)) at position [] and matcher [ ]

U3_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))
with rule U3_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))

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


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

(229) NO

(230) Obligation:

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

GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA

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

(231) UsableRulesProof (EQUIVALENT transformation)

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

(232) Obligation:

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

GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA

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

(233) PiDPToQDPProof (SOUND transformation)

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

(234) Obligation:

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

GT10_IN_AA → GT10_IN_AA

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

(235) 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 = GT10_IN_AA evaluates to t =GT10_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 GT10_IN_AA to GT10_IN_AA.

(236) NO

(237) Obligation:

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

QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)

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

(238) UsableRulesProof (EQUIVALENT transformation)

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

(239) Obligation:

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

QS80_IN_AA(.(T237, T231), X452) → U23_AA(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_AA(T235, X450)
QS80_IN_AA(.(T325, T231), X452) → U27_AA(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_AA(T325, T231, X452, T236, qs80_in_aa(T235, T319))
U28_AA(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, T324)
U27_AA(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → QS80_IN_AA(T235, T319)
U23_AA(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_AA(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_AA(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → QS80_IN_AA(T236, X451)

The TRS R consists of the following rules:

part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
qs80_in_aa([], []) → qs80_out_aa([], [])
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))

The argument filtering Pi contains the following mapping:
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
.(x1, x2)  =  .(x1, x2)
[]  =  []
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
QS80_IN_AA(x1, x2)  =  QS80_IN_AA
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U27_AA(x1, x2, x3, x4)  =  U27_AA(x4)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)

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

(240) PiDPToQDPProof (SOUND transformation)

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

(241) Obligation:

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

QS80_IN_AA → U23_AA(part87_in_aaaa)
U23_AA(part87_out_aaaa) → QS80_IN_AA
QS80_IN_AA → U27_AA(part87_in_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa)
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
app121_in_aaaa
U1_ga(x0)
U8_ga(x0)
U31_aaaa(x0)

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

(242) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS80_IN_AA → U23_AA(part87_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)

(243) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
QS80_IN_AA → U27_AA(part87_in_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa)
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
app121_in_aaaa
U1_ga(x0)
U8_ga(x0)
U31_aaaa(x0)

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

(244) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS80_IN_AA → U27_AA(part87_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)

(245) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa)
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)
QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
app121_in_aaaa
U1_ga(x0)
U8_ga(x0)
U31_aaaa(x0)

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

(246) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(part87_out_aaaa) → U28_AA(qs80_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U27_AA(part87_out_aaaa) → U28_AA(U22_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U23_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U27_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(qs80_out_aa)

(247) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa)
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)
QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(U22_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U23_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U27_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(qs80_out_aa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
app121_in_aaaa
U1_ga(x0)
U8_ga(x0)
U31_aaaa(x0)

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

(248) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U23_AA(part87_out_aaaa) → U25_AA(qs80_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U23_AA(part87_out_aaaa) → U25_AA(U22_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U23_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U27_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(qs80_out_aa)

(249) Obligation:

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

U23_AA(part87_out_aaaa) → QS80_IN_AA
U28_AA(qs80_out_aa) → QS80_IN_AA
U27_AA(part87_out_aaaa) → QS80_IN_AA
U25_AA(qs80_out_aa) → QS80_IN_AA
QS80_IN_AA → U23_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U23_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U23_AA(part87_out_aaaa)
QS80_IN_AA → U27_AA(U38_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U39_aaaa(gt10_in_aa))
QS80_IN_AA → U27_AA(U41_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(U42_aaaa(le100_in_aa))
QS80_IN_AA → U27_AA(part87_out_aaaa)
U27_AA(part87_out_aaaa) → U28_AA(U22_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U23_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(U27_aa(part87_in_aaaa))
U27_AA(part87_out_aaaa) → U28_AA(qs80_out_aa)
U23_AA(part87_out_aaaa) → U25_AA(U22_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U23_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(U27_aa(part87_in_aaaa))
U23_AA(part87_out_aaaa) → U25_AA(qs80_out_aa)

The TRS R consists of the following rules:

part87_in_aaaa → U38_aaaa(gt10_in_aa)
part87_in_aaaa → U39_aaaa(gt10_in_aa)
part87_in_aaaa → U41_aaaa(le100_in_aa)
part87_in_aaaa → U42_aaaa(le100_in_aa)
part87_in_aaaa → part87_out_aaaa
qs80_in_aa → U22_aa(part87_in_aaaa)
qs80_in_aa → U23_aa(part87_in_aaaa)
qs80_in_aa → U27_aa(part87_in_aaaa)
qs80_in_aa → qs80_out_aa
U38_aaaa(gt10_out_aa(T256, T257)) → part87_out_aaaa
U39_aaaa(gt10_out_aa(T261, T257)) → U40_aaaa(part20_in_gaaa(T261))
U41_aaaa(le100_out_aa(T283)) → part87_out_aaaa
U42_aaaa(le100_out_aa(T288)) → U43_aaaa(part20_in_gaaa(T288))
U22_aa(part87_out_aaaa) → qs80_out_aa
U23_aa(part87_out_aaaa) → U24_aa(qs80_in_aa)
U23_aa(part87_out_aaaa) → U25_aa(qs80_in_aa)
U27_aa(part87_out_aaaa) → U28_aa(qs80_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U40_aaaa(part20_out_gaaa) → part87_out_aaaa
le100_in_aa → U21_aa(le100_in_aa)
le100_in_aa → le100_out_aa(0)
U43_aaaa(part20_out_gaaa) → part87_out_aaaa
U24_aa(qs80_out_aa) → qs80_out_aa
U25_aa(qs80_out_aa) → U26_aa(qs80_in_aa)
U28_aa(qs80_out_aa) → U29_aa(qs80_in_aa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U21_aa(le100_out_aa(T302)) → le100_out_aa(s(T302))
U26_aa(qs80_out_aa) → qs80_out_aa
U29_aa(qs80_out_aa) → U30_aa(app121_in_aaaa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U30_aa(app121_out_aaaa) → qs80_out_aa
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
app121_in_aaaa → U31_aaaa(app121_in_aaaa)
app121_in_aaaa → app121_out_aaaa
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U31_aaaa(app121_out_aaaa) → app121_out_aaaa

The set Q consists of the following terms:

part87_in_aaaa
qs80_in_aa
U38_aaaa(x0)
U39_aaaa(x0)
U41_aaaa(x0)
U42_aaaa(x0)
U22_aa(x0)
U23_aa(x0)
U27_aa(x0)
gt10_in_aa
U40_aaaa(x0)
le100_in_aa
U43_aaaa(x0)
U24_aa(x0)
U25_aa(x0)
U28_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U21_aa(x0)
U26_aa(x0)
U29_aa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U30_aa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
app121_in_aaaa
U1_ga(x0)
U8_ga(x0)
U31_aaaa(x0)

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

(250) 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 = QS80_IN_AA evaluates to t =QS80_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS80_IN_AA → U23_AA(part87_out_aaaa)
with rule QS80_IN_AA → U23_AA(part87_out_aaaa) at position [] and matcher [ ]

U23_AA(part87_out_aaaa) → QS80_IN_AA
with rule U23_AA(part87_out_aaaa) → QS80_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.

(251) NO

(252) Obligation:

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

P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)

The TRS R consists of the following rules:

qs1_in_ag(.(T26, .(T27, T28)), T8) → U44_ag(T26, T27, T28, T8, gt10_in_aa(T26, T27))
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
U44_ag(T26, T27, T28, T8, gt10_out_aa(T26, T27)) → qs1_out_ag(.(T26, .(T27, T28)), T8)
qs1_in_ag(.(T31, .(T33, T32)), T8) → U45_ag(T31, T33, T32, T8, gt10_in_aa(T31, T33))
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U46_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
U46_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, X47, X48)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U45_ag(T31, T33, T32, T8, gt10_out_aa(T31, T33)) → U47_ag(T31, T33, T32, T8, part20_in_gaaa(T31, T32, T53, T54))
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U48_ag(T31, T33, T32, T8, qs51_in_gaa(T33, T53, X9))
qs51_in_gaa(T148, T149, X273) → U37_gaa(T148, T149, X273, p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273))
p56_in_gaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_gaaa(T148, T149, X269, X270))
U9_gaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_gaaa(T148, T149, X269, X270)) → p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U10_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
qs59_in_aa([], []) → qs59_out_aa([], [])
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
U11_gaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_gaaaaaa(T148, T149, T153, T154, X271, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_gaaa(T148, T149, T153, T154))
U12_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_gaaa(T148, T149, T153, T154)) → U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U14_gaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, X272, X273)
p56_in_gaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_gaaa(T148, T149, T153, T154))
U15_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_gaaa(T148, T149, T153, T154)) → U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
U17_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_agaa(T160, T148, T178, X273))
app70_in_agaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_agaa(T201, T202, T203, T204, X391, app70_in_agaa(T202, T203, T204, X391))
app70_in_agaa([], T213, T214, .(T213, T214)) → app70_out_agaa([], T213, T214, .(T213, T214))
U20_agaa(T201, T202, T203, T204, X391, app70_out_agaa(T202, T203, T204, X391)) → app70_out_agaa(.(T201, T202), T203, T204, .(T201, X391))
U18_gaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_agaa(T160, T148, T178, X273)) → p56_out_gaaaaaa(T148, T149, T153, T154, T160, T178, X273)
U37_gaa(T148, T149, X273, p56_out_gaaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qs51_out_gaa(T148, T149, X273)
U48_ag(T31, T33, T32, T8, qs51_out_gaa(T33, T53, X9)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
U47_ag(T31, T33, T32, T8, part20_out_gaaa(T31, T32, T53, T54)) → U49_ag(T31, T33, T32, T8, T54, qs51_in_gaa(T33, T53, T131))
U49_ag(T31, T33, T32, T8, T54, qs51_out_gaa(T33, T53, T131)) → U50_ag(T31, T33, T32, T8, p52_in_aaagg(T54, X10, T131, T31, T8))
p52_in_aaagg(T54, X10, T131, T31, T8) → U33_aaagg(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
qs80_in_aa(.(T230, T231), X452) → U22_aa(T230, T231, X452, part87_in_aaaa(T230, T231, X448, X449))
part87_in_aaaa(T256, .(T257, T258), .(T257, X503), X504) → U38_aaaa(T256, T257, T258, X503, X504, gt10_in_aa(T256, T257))
U38_aaaa(T256, T257, T258, X503, X504, gt10_out_aa(T256, T257)) → part87_out_aaaa(T256, .(T257, T258), .(T257, X503), X504)
part87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U39_aaaa(T261, T257, T262, X503, X504, gt10_in_aa(T261, T257))
U39_aaaa(T261, T257, T262, X503, X504, gt10_out_aa(T261, T257)) → U40_aaaa(T261, T257, T262, X503, X504, part20_in_gaaa(T261, T262, X503, X504))
U40_aaaa(T261, T257, T262, X503, X504, part20_out_gaaa(T261, T262, X503, X504)) → part87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
part87_in_aaaa(T283, .(T284, T285), X554, .(T284, X555)) → U41_aaaa(T283, T284, T285, X554, X555, le100_in_aa(T283, T284))
le100_in_aa(s(T302), s(T303)) → U21_aa(T302, T303, le100_in_aa(T302, T303))
le100_in_aa(0, s(T310)) → le100_out_aa(0, s(T310))
le100_in_aa(0, 0) → le100_out_aa(0, 0)
U21_aa(T302, T303, le100_out_aa(T302, T303)) → le100_out_aa(s(T302), s(T303))
U41_aaaa(T283, T284, T285, X554, X555, le100_out_aa(T283, T284)) → part87_out_aaaa(T283, .(T284, T285), X554, .(T284, X555))
part87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U42_aaaa(T288, T284, T289, X554, X555, le100_in_aa(T288, T284))
U42_aaaa(T288, T284, T289, X554, X555, le100_out_aa(T288, T284)) → U43_aaaa(T288, T284, T289, X554, X555, part20_in_gaaa(T288, T289, X554, X555))
U43_aaaa(T288, T284, T289, X554, X555, part20_out_gaaa(T288, T289, X554, X555)) → part87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
part87_in_aaaa(T316, [], [], []) → part87_out_aaaa(T316, [], [], [])
U22_aa(T230, T231, X452, part87_out_aaaa(T230, T231, X448, X449)) → qs80_out_aa(.(T230, T231), X452)
qs80_in_aa(.(T237, T231), X452) → U23_aa(T237, T231, X452, part87_in_aaaa(T237, T231, T235, T236))
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U24_aa(T237, T231, X452, qs80_in_aa(T235, X450))
qs80_in_aa(.(T325, T231), X452) → U27_aa(T325, T231, X452, part87_in_aaaa(T325, T231, T235, T236))
U27_aa(T325, T231, X452, part87_out_aaaa(T325, T231, T235, T236)) → U28_aa(T325, T231, X452, T236, qs80_in_aa(T235, T319))
qs80_in_aa([], []) → qs80_out_aa([], [])
U28_aa(T325, T231, X452, T236, qs80_out_aa(T235, T319)) → U29_aa(T325, T231, X452, T319, qs80_in_aa(T236, T324))
U29_aa(T325, T231, X452, T319, qs80_out_aa(T236, T324)) → U30_aa(T325, T231, X452, app121_in_aaaa(T319, T325, T324, X452))
app121_in_aaaa(.(T348, T349), T352, T353, .(T348, X657)) → U31_aaaa(T348, T349, T352, T353, X657, app121_in_aaaa(T349, T352, T353, X657))
app121_in_aaaa([], T362, T363, .(T362, T363)) → app121_out_aaaa([], T362, T363, .(T362, T363))
U31_aaaa(T348, T349, T352, T353, X657, app121_out_aaaa(T349, T352, T353, X657)) → app121_out_aaaa(.(T348, T349), T352, T353, .(T348, X657))
U30_aa(T325, T231, X452, app121_out_aaaa(T319, T325, T324, X452)) → qs80_out_aa(.(T325, T231), X452)
U24_aa(T237, T231, X452, qs80_out_aa(T235, X450)) → qs80_out_aa(.(T237, T231), X452)
U23_aa(T237, T231, X452, part87_out_aaaa(T237, T231, T235, T236)) → U25_aa(T237, T231, X452, T236, qs80_in_aa(T235, T319))
U25_aa(T237, T231, X452, T236, qs80_out_aa(T235, T319)) → U26_aa(T237, T231, X452, qs80_in_aa(T236, X451))
U26_aa(T237, T231, X452, qs80_out_aa(T236, X451)) → qs80_out_aa(.(T237, T231), X452)
U33_aaagg(T54, X10, T131, T31, T8, qs80_out_aa(T54, X10)) → p52_out_aaagg(T54, X10, T131, T31, T8)
p52_in_aaagg(T54, T217, T131, T31, T8) → U34_aaagg(T54, T217, T131, T31, T8, qs80_in_aa(T54, T217))
U34_aaagg(T54, T217, T131, T31, T8, qs80_out_aa(T54, T217)) → U35_aaagg(T54, T217, T131, T31, T8, app81_in_agag(T131, T31, T217, T8))
app81_in_agag(.(T389, T390), T391, T394, .(T389, T393)) → U32_agag(T389, T390, T391, T394, T393, app81_in_agag(T390, T391, T394, T393))
app81_in_agag([], T404, T405, .(T404, T405)) → app81_out_agag([], T404, T405, .(T404, T405))
U32_agag(T389, T390, T391, T394, T393, app81_out_agag(T390, T391, T394, T393)) → app81_out_agag(.(T389, T390), T391, T394, .(T389, T393))
U35_aaagg(T54, T217, T131, T31, T8, app81_out_agag(T131, T31, T217, T8)) → p52_out_aaagg(T54, T217, T131, T31, T8)
U50_ag(T31, T33, T32, T8, p52_out_aaagg(T54, X10, T131, T31, T8)) → qs1_out_ag(.(T31, .(T33, T32)), T8)
qs1_in_ag(.(T421, .(T422, T423)), T8) → U51_ag(T421, T422, T423, T8, le100_in_aa(T421, T422))
U51_ag(T421, T422, T423, T8, le100_out_aa(T421, T422)) → qs1_out_ag(.(T421, .(T422, T423)), T8)
qs1_in_ag(.(T426, .(T428, T427)), T8) → U52_ag(T426, T428, T427, T8, le100_in_aa(T426, T428))
U52_ag(T426, T428, T427, T8, le100_out_aa(T426, T428)) → U53_ag(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U53_ag(T426, T428, T427, T8, part20_out_gaaa(T426, T427, X740, X741)) → qs1_out_ag(.(T426, .(T428, T427)), T8)
qs1_in_ag(.(T426, .(T436, T427)), T8) → U54_ag(T426, T436, T427, T8, le100_in_aa(T426, T436))
U54_ag(T426, T436, T427, T8, le100_out_aa(T426, T436)) → U55_ag(T426, T436, T427, T8, part20_in_gaaa(T426, T427, T434, T435))
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U56_ag(T426, T436, T427, T8, qs80_in_aa(T434, X9))
U56_ag(T426, T436, T427, T8, qs80_out_aa(T434, X9)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
U55_ag(T426, T436, T427, T8, part20_out_gaaa(T426, T427, T434, T435)) → U57_ag(T426, T436, T427, T8, T435, qs80_in_aa(T434, T442))
U57_ag(T426, T436, T427, T8, T435, qs80_out_aa(T434, T442)) → U58_ag(T426, T436, T427, T8, p52_in_aaagg(.(T436, T435), X10, T442, T426, T8))
U58_ag(T426, T436, T427, T8, p52_out_aaagg(.(T436, T435), X10, T442, T426, T8)) → qs1_out_ag(.(T426, .(T436, T427)), T8)
qs1_in_ag(.(T452, []), T8) → U59_ag(T452, T8, qs155_in_a(X9))
qs155_in_a([]) → qs155_out_a([])
U59_ag(T452, T8, qs155_out_a(X9)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag(.(T452, []), T8) → U60_ag(T452, T8, qs155_in_a(T455))
U60_ag(T452, T8, qs155_out_a(T455)) → U61_ag(T452, T8, qs155_in_a(X10))
U61_ag(T452, T8, qs155_out_a(X10)) → qs1_out_ag(.(T452, []), T8)
U60_ag(T452, T8, qs155_out_a(T455)) → U62_ag(T452, T8, T455, qs155_in_a(T460))
U62_ag(T452, T8, T455, qs155_out_a(T460)) → U63_ag(T452, T8, app162_in_gagg(T455, T452, T460, T8))
app162_in_gagg(.(T486, T487), T491, T489, .(T486, T490)) → U36_gagg(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
app162_in_gagg([], T501, T502, .(T501, T502)) → app162_out_gagg([], T501, T502, .(T501, T502))
U36_gagg(T486, T487, T491, T489, T490, app162_out_gagg(T487, T491, T489, T490)) → app162_out_gagg(.(T486, T487), T491, T489, .(T486, T490))
U63_ag(T452, T8, app162_out_gagg(T455, T452, T460, T8)) → qs1_out_ag(.(T452, []), T8)
qs1_in_ag([], []) → qs1_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ag(x1, x2)  =  qs1_in_ag(x2)
U44_ag(x1, x2, x3, x4, x5)  =  U44_ag(x5)
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
qs1_out_ag(x1, x2)  =  qs1_out_ag
U45_ag(x1, x2, x3, x4, x5)  =  U45_ag(x4, x5)
U46_ag(x1, x2, x3, x4, x5)  =  U46_ag(x5)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
U47_ag(x1, x2, x3, x4, x5)  =  U47_ag(x1, x2, x4, x5)
U48_ag(x1, x2, x3, x4, x5)  =  U48_ag(x5)
qs51_in_gaa(x1, x2, x3)  =  qs51_in_gaa(x1)
U37_gaa(x1, x2, x3, x4)  =  U37_gaa(x4)
p56_in_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_gaaaaaa(x1)
U9_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_gaaaaaa(x8)
p56_out_gaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_gaaaaaa
U10_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_gaaaaaa(x8)
U11_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gaaaaaa(x8)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
U12_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_gaaaaaa(x8)
U13_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_gaaaaaa(x8)
U14_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_gaaaaaa(x8)
U15_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_gaaaaaa(x1, x8)
U16_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_gaaaaaa(x1, x8)
U17_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_gaaaaaa(x1, x8)
U18_gaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_gaaaaaa(x8)
app70_in_agaa(x1, x2, x3, x4)  =  app70_in_agaa(x2)
U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x6)
app70_out_agaa(x1, x2, x3, x4)  =  app70_out_agaa
qs51_out_gaa(x1, x2, x3)  =  qs51_out_gaa
U49_ag(x1, x2, x3, x4, x5, x6)  =  U49_ag(x1, x4, x6)
U50_ag(x1, x2, x3, x4, x5)  =  U50_ag(x5)
p52_in_aaagg(x1, x2, x3, x4, x5)  =  p52_in_aaagg(x4, x5)
U33_aaagg(x1, x2, x3, x4, x5, x6)  =  U33_aaagg(x6)
qs80_in_aa(x1, x2)  =  qs80_in_aa
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
part87_in_aaaa(x1, x2, x3, x4)  =  part87_in_aaaa
U38_aaaa(x1, x2, x3, x4, x5, x6)  =  U38_aaaa(x6)
part87_out_aaaa(x1, x2, x3, x4)  =  part87_out_aaaa
U39_aaaa(x1, x2, x3, x4, x5, x6)  =  U39_aaaa(x6)
U40_aaaa(x1, x2, x3, x4, x5, x6)  =  U40_aaaa(x6)
U41_aaaa(x1, x2, x3, x4, x5, x6)  =  U41_aaaa(x6)
le100_in_aa(x1, x2)  =  le100_in_aa
U21_aa(x1, x2, x3)  =  U21_aa(x3)
le100_out_aa(x1, x2)  =  le100_out_aa(x1)
U42_aaaa(x1, x2, x3, x4, x5, x6)  =  U42_aaaa(x6)
U43_aaaa(x1, x2, x3, x4, x5, x6)  =  U43_aaaa(x6)
qs80_out_aa(x1, x2)  =  qs80_out_aa
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U27_aa(x1, x2, x3, x4)  =  U27_aa(x4)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
app121_in_aaaa(x1, x2, x3, x4)  =  app121_in_aaaa
U31_aaaa(x1, x2, x3, x4, x5, x6)  =  U31_aaaa(x6)
app121_out_aaaa(x1, x2, x3, x4)  =  app121_out_aaaa
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
p52_out_aaagg(x1, x2, x3, x4, x5)  =  p52_out_aaagg
U34_aaagg(x1, x2, x3, x4, x5, x6)  =  U34_aaagg(x4, x5, x6)
U35_aaagg(x1, x2, x3, x4, x5, x6)  =  U35_aaagg(x6)
app81_in_agag(x1, x2, x3, x4)  =  app81_in_agag(x2, x4)
U32_agag(x1, x2, x3, x4, x5, x6)  =  U32_agag(x1, x6)
app81_out_agag(x1, x2, x3, x4)  =  app81_out_agag(x1, x3)
U51_ag(x1, x2, x3, x4, x5)  =  U51_ag(x5)
U52_ag(x1, x2, x3, x4, x5)  =  U52_ag(x5)
U53_ag(x1, x2, x3, x4, x5)  =  U53_ag(x5)
U54_ag(x1, x2, x3, x4, x5)  =  U54_ag(x4, x5)
U55_ag(x1, x2, x3, x4, x5)  =  U55_ag(x1, x4, x5)
U56_ag(x1, x2, x3, x4, x5)  =  U56_ag(x5)
U57_ag(x1, x2, x3, x4, x5, x6)  =  U57_ag(x1, x4, x6)
U58_ag(x1, x2, x3, x4, x5)  =  U58_ag(x5)
U59_ag(x1, x2, x3)  =  U59_ag(x3)
qs155_in_a(x1)  =  qs155_in_a
qs155_out_a(x1)  =  qs155_out_a(x1)
U60_ag(x1, x2, x3)  =  U60_ag(x2, x3)
U61_ag(x1, x2, x3)  =  U61_ag(x3)
U62_ag(x1, x2, x3, x4)  =  U62_ag(x2, x3, x4)
U63_ag(x1, x2, x3)  =  U63_ag(x3)
app162_in_gagg(x1, x2, x3, x4)  =  app162_in_gagg(x1, x3, x4)
U36_gagg(x1, x2, x3, x4, x5, x6)  =  U36_gagg(x6)
app162_out_gagg(x1, x2, x3, x4)  =  app162_out_gagg(x2)
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)

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

(253) UsableRulesProof (EQUIVALENT transformation)

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

(254) Obligation:

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

P56_IN_AAAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U10_AAAAAAA(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, X271)
QS59_IN_AA(.(T171, T172), X335) → P56_IN_AAAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U13_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, X272)
U12_AAAAAAA(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)
P56_IN_AAAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U16_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → QS59_IN_AA(T154, T178)
U15_AAAAAAA(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → QS59_IN_AA(T153, T160)

The TRS R consists of the following rules:

part20_in_aaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_aaaa(T70, T73, T74, X119, X120, gt10_in_aa(T70, T73))
part20_in_aaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_aaaa(T70, T73, T77, X119, X120, gt10_in_aa(T70, T73))
part20_in_aaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_aaaa(T95, T98, T99, X170, X171, le33_in_aa(T95, T98))
part20_in_aaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_aaaa(T95, T98, T102, X170, X171, le33_in_aa(T95, T98))
part20_in_aaaa(T128, [], [], []) → part20_out_aaaa(T128, [], [], [])
qs59_in_aa(.(T171, T172), X335) → U19_aa(T171, T172, X335, p56_in_aaaaaaa(T171, T172, X331, X332, X333, X334, X335))
qs59_in_aa([], []) → qs59_out_aa([], [])
U2_aaaa(T70, T73, T74, X119, X120, gt10_out_aa(T70, T73)) → part20_out_aaaa(T70, .(T73, T74), .(T73, X119), X120)
U3_aaaa(T70, T73, T77, X119, X120, gt10_out_aa(T70, T73)) → U4_aaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U5_aaaa(T95, T98, T99, X170, X171, le33_out_aa(T95, T98)) → part20_out_aaaa(T95, .(T98, T99), X170, .(T98, X171))
U6_aaaa(T95, T98, T102, X170, X171, le33_out_aa(T95, T98)) → U7_aaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U19_aa(T171, T172, X335, p56_out_aaaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qs59_out_aa(.(T171, T172), X335)
gt10_in_aa(s(T46), s(T47)) → U1_aa(T46, T47, gt10_in_aa(T46, T47))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U4_aaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_aaaa(T70, .(T73, T77), .(T73, X119), X120)
le33_in_aa(s(T113), s(T115)) → U8_aa(T113, T115, le33_in_aa(T113, T115))
le33_in_aa(0, s(T122)) → le33_out_aa(0, s(T122))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U7_aaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_aaaa(T95, .(T98, T102), X170, .(T98, X171))
p56_in_aaaaaaa(T148, T149, X269, X270, X271, X272, X273) → U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_in_aaaa(T148, T149, X269, X270))
p56_in_aaaaaaa(T148, T149, T153, T154, X271, X272, X273) → U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
p56_in_aaaaaaa(T148, T149, T153, T154, T160, X272, X273) → U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_in_aaaa(T148, T149, T153, T154))
p56_in_aaaaaaa(T148, T149, T153, T154, T160, T178, X273) → U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_in_aaaa(T148, T149, T153, T154))
U1_aa(T46, T47, gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70, .(T73, T74), .(T73, X119), X120) → U2_gaaa(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U3_gaaa(T70, T73, T77, X119, X120, gt10_in_ga(T70, T73))
part20_in_gaaa(T95, .(T98, T99), X170, .(T98, X171)) → U5_gaaa(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U6_gaaa(T95, T98, T102, X170, X171, le33_in_ga(T95, T98))
part20_in_gaaa(T128, [], [], []) → part20_out_gaaa(T128, [], [], [])
U8_aa(T113, T115, le33_out_aa(T113, T115)) → le33_out_aa(s(T113), s(T115))
U9_aaaaaaa(T148, T149, X269, X270, X271, X272, X273, part20_out_aaaa(T148, T149, X269, X270)) → p56_out_aaaaaaa(T148, T149, X269, X270, X271, X272, X273)
U10_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_in_aa(T153, X271))
U12_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, part20_out_aaaa(T148, T149, T153, T154)) → U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T153, T160))
U15_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, part20_out_aaaa(T148, T149, T153, T154)) → U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T153, T160))
U2_gaaa(T70, T73, T74, X119, X120, gt10_out_ga(T70, T73)) → part20_out_gaaa(T70, .(T73, T74), .(T73, X119), X120)
U3_gaaa(T70, T73, T77, X119, X120, gt10_out_ga(T70, T73)) → U4_gaaa(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U5_gaaa(T95, T98, T99, X170, X171, le33_out_ga(T95, T98)) → part20_out_gaaa(T95, .(T98, T99), X170, .(T98, X171))
U6_gaaa(T95, T98, T102, X170, X171, le33_out_ga(T95, T98)) → U7_gaaa(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U11_aaaaaaa(T148, T149, T153, T154, X271, X272, X273, qs59_out_aa(T153, X271)) → p56_out_aaaaaaa(T148, T149, T153, T154, X271, X272, X273)
U13_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T153, T160)) → U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_in_aa(T154, X272))
U16_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T153, T160)) → U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_in_aa(T154, T178))
gt10_in_ga(s(T46), s(T47)) → U1_ga(T46, T47, gt10_in_ga(T46, T47))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U4_gaaa(T70, T73, T77, X119, X120, part20_out_gaaa(T70, T77, X119, X120)) → part20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
le33_in_ga(s(T113), s(T115)) → U8_ga(T113, T115, le33_in_ga(T113, T115))
le33_in_ga(0, s(T122)) → le33_out_ga(0, s(T122))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U7_gaaa(T95, T98, T102, X170, X171, part20_out_gaaa(T95, T102, X170, X171)) → part20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U14_aaaaaaa(T148, T149, T153, T154, T160, X272, X273, qs59_out_aa(T154, X272)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, X272, X273)
U17_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, qs59_out_aa(T154, T178)) → U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_in_aaaa(T160, T148, T178, X273))
U1_ga(T46, T47, gt10_out_ga(T46, T47)) → gt10_out_ga(s(T46), s(T47))
U8_ga(T113, T115, le33_out_ga(T113, T115)) → le33_out_ga(s(T113), s(T115))
U18_aaaaaaa(T148, T149, T153, T154, T160, T178, X273, app70_out_aaaa(T160, T148, T178, X273)) → p56_out_aaaaaaa(T148, T149, T153, T154, T160, T178, X273)
app70_in_aaaa(.(T201, T202), T203, T204, .(T201, X391)) → U20_aaaa(T201, T202, T203, T204, X391, app70_in_aaaa(T202, T203, T204, X391))
app70_in_aaaa([], T213, T214, .(T213, T214)) → app70_out_aaaa([], T213, T214, .(T213, T214))
U20_aaaa(T201, T202, T203, T204, X391, app70_out_aaaa(T202, T203, T204, X391)) → app70_out_aaaa(.(T201, T202), T203, T204, .(T201, X391))

The argument filtering Pi contains the following mapping:
gt10_in_aa(x1, x2)  =  gt10_in_aa
U1_aa(x1, x2, x3)  =  U1_aa(x3)
gt10_out_aa(x1, x2)  =  gt10_out_aa(x1, x2)
part20_in_gaaa(x1, x2, x3, x4)  =  part20_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6)  =  U2_gaaa(x6)
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
0  =  0
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
part20_out_gaaa(x1, x2, x3, x4)  =  part20_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5, x6)  =  U3_gaaa(x1, x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x6)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
qs59_in_aa(x1, x2)  =  qs59_in_aa
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
p56_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_in_aaaaaaa
U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_aaaaaaa(x8)
part20_in_aaaa(x1, x2, x3, x4)  =  part20_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5, x6)  =  U2_aaaa(x6)
part20_out_aaaa(x1, x2, x3, x4)  =  part20_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U8_aa(x1, x2, x3)  =  U8_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
p56_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  p56_out_aaaaaaa
U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_aaaaaaa(x8)
U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_aaaaaaa(x8)
qs59_out_aa(x1, x2)  =  qs59_out_aa
U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_aaaaaaa(x8)
U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_aaaaaaa(x8)
U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_aaaaaaa(x8)
U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_aaaaaaa(x8)
U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_aaaaaaa(x8)
U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U17_aaaaaaa(x8)
U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaaaaa(x8)
.(x1, x2)  =  .(x1, x2)
[]  =  []
app70_in_aaaa(x1, x2, x3, x4)  =  app70_in_aaaa
U20_aaaa(x1, x2, x3, x4, x5, x6)  =  U20_aaaa(x6)
app70_out_aaaa(x1, x2, x3, x4)  =  app70_out_aaaa
QS59_IN_AA(x1, x2)  =  QS59_IN_AA
P56_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  P56_IN_AAAAAAA
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_AAAAAAA(x8)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_AAAAAAA(x8)

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

(255) PiDPToQDPProof (SOUND transformation)

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

(256) Obligation:

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

P56_IN_AAAAAAA → U10_AAAAAAA(part20_in_aaaa)
U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
P56_IN_AAAAAAA → U12_AAAAAAA(part20_in_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa)
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0)
U8_ga(x0)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(257) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P56_IN_AAAAAAA → U10_AAAAAAA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)

(258) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
P56_IN_AAAAAAA → U12_AAAAAAA(part20_in_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa)
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0)
U8_ga(x0)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(259) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P56_IN_AAAAAAA → U12_AAAAAAA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)

(260) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa)
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0)
U8_ga(x0)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(261) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)

(262) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0)
U8_ga(x0)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(263) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P56_IN_AAAAAAA → U15_AAAAAAA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P56_IN_AAAAAAA → U15_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(part20_out_aaaa)

(264) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa)
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)
P56_IN_AAAAAAA → U15_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0)
U8_ga(x0)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(265) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_out_aa)

(266) Obligation:

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

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
QS59_IN_AA → P56_IN_AAAAAAA
U13_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U12_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
U16_AAAAAAA(qs59_out_aa) → QS59_IN_AA
U15_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
P56_IN_AAAAAAA → U10_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
P56_IN_AAAAAAA → U12_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U12_AAAAAAA(part20_out_aaaa)
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U12_AAAAAAA(part20_out_aaaa) → U13_AAAAAAA(qs59_out_aa)
P56_IN_AAAAAAA → U15_AAAAAAA(U2_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U3_aaaa(gt10_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U5_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(U6_aaaa(le33_in_aa))
P56_IN_AAAAAAA → U15_AAAAAAA(part20_out_aaaa)
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(U19_aa(p56_in_aaaaaaa))
U15_AAAAAAA(part20_out_aaaa) → U16_AAAAAAA(qs59_out_aa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U3_aaaa(gt10_in_aa)
part20_in_aaaa → U5_aaaa(le33_in_aa)
part20_in_aaaa → U6_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs59_in_aa → U19_aa(p56_in_aaaaaaa)
qs59_in_aa → qs59_out_aa
U2_aaaa(gt10_out_aa(T70, T73)) → part20_out_aaaa
U3_aaaa(gt10_out_aa(T70, T73)) → U4_aaaa(part20_in_gaaa(T70))
U5_aaaa(le33_out_aa(T95)) → part20_out_aaaa
U6_aaaa(le33_out_aa(T95)) → U7_aaaa(part20_in_gaaa(T95))
U19_aa(p56_out_aaaaaaa) → qs59_out_aa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U4_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U8_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U7_aaaa(part20_out_gaaa) → part20_out_aaaa
p56_in_aaaaaaa → U9_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U10_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U12_aaaaaaa(part20_in_aaaa)
p56_in_aaaaaaa → U15_aaaaaaa(part20_in_aaaa)
U1_aa(gt10_out_aa(T46, T47)) → gt10_out_aa(s(T46), s(T47))
part20_in_gaaa(T70) → U2_gaaa(gt10_in_ga(T70))
part20_in_gaaa(T70) → U3_gaaa(T70, gt10_in_ga(T70))
part20_in_gaaa(T95) → U5_gaaa(le33_in_ga(T95))
part20_in_gaaa(T95) → U6_gaaa(T95, le33_in_ga(T95))
part20_in_gaaa(T128) → part20_out_gaaa
U8_aa(le33_out_aa(T113)) → le33_out_aa(s(T113))
U9_aaaaaaa(part20_out_aaaa) → p56_out_aaaaaaa
U10_aaaaaaa(part20_out_aaaa) → U11_aaaaaaa(qs59_in_aa)
U12_aaaaaaa(part20_out_aaaa) → U13_aaaaaaa(qs59_in_aa)
U15_aaaaaaa(part20_out_aaaa) → U16_aaaaaaa(qs59_in_aa)
U2_gaaa(gt10_out_ga(T73)) → part20_out_gaaa
U3_gaaa(T70, gt10_out_ga(T73)) → U4_gaaa(part20_in_gaaa(T70))
U5_gaaa(le33_out_ga) → part20_out_gaaa
U6_gaaa(T95, le33_out_ga) → U7_gaaa(part20_in_gaaa(T95))
U11_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U13_aaaaaaa(qs59_out_aa) → U14_aaaaaaa(qs59_in_aa)
U16_aaaaaaa(qs59_out_aa) → U17_aaaaaaa(qs59_in_aa)
gt10_in_ga(s(T46)) → U1_ga(gt10_in_ga(T46))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U4_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T113)) → U8_ga(le33_in_ga(T113))
le33_in_ga(0) → le33_out_ga
U7_gaaa(part20_out_gaaa) → part20_out_gaaa
U14_aaaaaaa(qs59_out_aa) → p56_out_aaaaaaa
U17_aaaaaaa(qs59_out_aa) → U18_aaaaaaa(app70_in_aaaa)
U1_ga(gt10_out_ga(T47)) → gt10_out_ga(s(T47))
U8_ga(le33_out_ga) → le33_out_ga
U18_aaaaaaa(app70_out_aaaa) → p56_out_aaaaaaa
app70_in_aaaa → U20_aaaa(app70_in_aaaa)
app70_in_aaaa → app70_out_aaaa
U20_aaaa(app70_out_aaaa) → app70_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs59_in_aa
U2_aaaa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U6_aaaa(x0)
U19_aa(x0)
gt10_in_aa
U4_aaaa(x0)
le33_in_aa
U7_aaaa(x0)
p56_in_aaaaaaa
U1_aa(x0)
part20_in_gaaa(x0)
U8_aa(x0)
U9_aaaaaaa(x0)
U10_aaaaaaa(x0)
U12_aaaaaaa(x0)
U15_aaaaaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0, x1)
U5_gaaa(x0)
U6_gaaa(x0, x1)
U11_aaaaaaa(x0)
U13_aaaaaaa(x0)
U16_aaaaaaa(x0)
gt10_in_ga(x0)
U4_gaaa(x0)
le33_in_ga(x0)
U7_gaaa(x0)
U14_aaaaaaa(x0)
U17_aaaaaaa(x0)
U1_ga(x0)
U8_ga(x0)
U18_aaaaaaa(x0)
app70_in_aaaa
U20_aaaa(x0)

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

(267) 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 = QS59_IN_AA evaluates to t =QS59_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

QS59_IN_AA → P56_IN_AAAAAAA
with rule QS59_IN_AA → P56_IN_AAAAAAA at position [] and matcher [ ]

P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa)
with rule P56_IN_AAAAAAA → U10_AAAAAAA(part20_out_aaaa) at position [] and matcher [ ]

U10_AAAAAAA(part20_out_aaaa) → QS59_IN_AA
with rule U10_AAAAAAA(part20_out_aaaa) → QS59_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.

(268) NO