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

(0) Obligation:

Clauses:

qs(.(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(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: qs(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: qs(T1, T2), SCOPE: 1, CLAUSE: 0 | qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))) | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 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\nT1 is ground\nqs(T1, T2) !=? qs(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 2 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(gt(T22, T23), ','(part(T22, T24, X47, X48), ','(qs(.(T23, X47), X9), ','(qs(X48, X10), app(X9, .(T22, X10), T9)))))\n\nT22 is ground\nT23 is ground\nT24 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(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(part(T22, T24, X47, X48), ','(qs(.(T23, X47), X9), ','(qs(X48, X10), app(X9, .(T22, X10), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX9 is free; \nX10 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: gt(T22, T23), SCOPE: 3, CLAUSE: 7 | gt(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: gt(T22, T23), SCOPE: 3, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: gt(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: gt(T37, T38)\n\nT37 is ground\nT38 is ground", 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(T22, T24, X47, X48)\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(qs(.(T23, T44), X9), ','(qs(T45, X10), app(X9, .(T22, X10), T9)))\n\nT22 is ground\nT23 is ground\nT44 is ground\nT45 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 2 | part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(gt(T61, T62), part(T61, T63, X119, X120))\n\nT61 is ground\nT62 is ground\nT63 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(T61, T62)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: part(T61, T63, X119, X120)\n\nT61 is ground\nT63 is ground\nX119 is free; \nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 3\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(le(T83, T84), part(T83, T85, X170, X171))\n\nT83 is ground\nT84 is ground\nT85 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(T83, T84)\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T83, T85, X170, X171)\n\nT83 is ground\nT85 is ground\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: le(T83, T84), SCOPE: 5, CLAUSE: 9 | le(T83, T84), SCOPE: 5, CLAUSE: 10 | le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T83, T84), SCOPE: 5, CLAUSE: 9\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T83, T84), SCOPE: 5, CLAUSE: 10 | le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T98, T99)\n\nT98 is ground\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: le(T83, T84), SCOPE: 5, CLAUSE: 10\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 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(.(T23, T44), X9)\n\nT23 is ground\nT44 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(qs(T45, X10), app(T115, .(T22, X10), T9))\n\nT22 is ground\nT45 is ground\nT115 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: qs(T45, X10)\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: app(T115, .(T22, T120), T9)\n\nT22 is ground\nT115 is ground\nT120 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: qs(T45, X10), SCOPE: 6, CLAUSE: 0 | qs(T45, X10), SCOPE: 6, CLAUSE: 1\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: qs(T45, X10), SCOPE: 6, CLAUSE: 0\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: qs(T45, X10), SCOPE: 6, CLAUSE: 1\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(part(T131, T132, X271, X272), ','(qs(X271, X273), ','(qs(X272, X274), app(X273, .(T131, X274), X275))))\n\nT131 is ground\nT132 is ground\nX275 is free; \nX271 is free; \nX272 is free; \nX273 is free; \nX274 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: part(T131, T132, X271, X272)\n\nT131 is ground\nT132 is ground\nX271 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(qs(T136, X273), ','(qs(T137, X274), app(X273, .(T131, X274), X275)))\n\nT131 is ground\nT136 is ground\nT137 is ground\nX275 is free; \nX273 is free; \nX274 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: qs(T136, X273)\n\nT136 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(qs(T137, X274), app(T143, .(T131, X274), X275))\n\nT131 is ground\nT137 is ground\nT143 is ground\nX275 is free; \nX274 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: qs(T137, X274)\n\nT137 is ground\nX274 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: app(T143, .(T131, T148), X275)\n\nT131 is ground\nT143 is ground\nT148 is ground\nX275 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: app(T143, .(T131, T148), X275), SCOPE: 7, CLAUSE: 5 | app(T143, .(T131, T148), X275), SCOPE: 7, CLAUSE: 6\n\nT131 is ground\nT143 is ground\nT148 is ground\nX275 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app(T143, .(T131, T148), X275), SCOPE: 7, CLAUSE: 5\n\nT131 is ground\nT143 is ground\nT148 is ground\nX275 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: app(T143, .(T131, T148), X275), SCOPE: 7, CLAUSE: 6\n\nT131 is ground\nT143 is ground\nT148 is ground\nX275 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: app(T172, .(T173, T174), X354)\n\nT172 is ground\nT173 is ground\nT174 is ground\nX354 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: app(T115, .(T22, T120), T9), SCOPE: 8, CLAUSE: 5 | app(T115, .(T22, T120), T9), SCOPE: 8, CLAUSE: 6\n\nT22 is ground\nT115 is ground\nT120 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: app(T115, .(T22, T120), T9), SCOPE: 8, CLAUSE: 5\n\nT22 is ground\nT115 is ground\nT120 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app(T115, .(T22, T120), T9), SCOPE: 8, CLAUSE: 6\n\nT22 is ground\nT115 is ground\nT120 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: app(T211, .(T212, T213), T215)\n\nT211 is ground\nT212 is ground\nT213 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(le(T239, T240), ','(part(T239, T241, X437, X438), ','(qs(X437, X9), ','(qs(.(T240, X438), X10), app(X9, .(T239, X10), T9)))))\n\nT239 is ground\nT240 is ground\nT241 is ground\nX9 is free; \nX10 is free; \nX437 is free; \nX438 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: le(T239, T240)\n\nT239 is ground\nT240 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(part(T239, T241, X437, X438), ','(qs(X437, X9), ','(qs(.(T240, X438), X10), app(X9, .(T239, X10), T9))))\n\nT239 is ground\nT240 is ground\nT241 is ground\nX9 is free; \nX10 is free; \nX437 is free; \nX438 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: part(T239, T241, X437, X438)\n\nT239 is ground\nT241 is ground\nX437 is free; \nX438 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(qs(T249, X9), ','(qs(.(T240, T250), X10), app(X9, .(T239, X10), T9)))\n\nT239 is ground\nT240 is ground\nT249 is ground\nT250 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: qs(T249, X9)\n\nT249 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(qs(.(T240, T250), X10), app(T256, .(T239, X10), T9))\n\nT239 is ground\nT240 is ground\nT250 is ground\nT256 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(qs([], X9), ','(qs([], X10), app(X9, .(T265, X10), T9)))\n\nT265 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: qs([], X9)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(qs([], X10), app(T268, .(T265, X10), T9))\n\nT265 is ground\nT268 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: qs([], X9), SCOPE: 9, CLAUSE: 0 | qs([], X9), SCOPE: 9, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: qs([], X9), SCOPE: 9, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: qs([], X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: app(T268, .(T265, T273), T9)\n\nT265 is ground\nT268 is ground\nT273 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 112 [arrowhead = none ];
112 -> 2;

113 [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 -> T6\nX5 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX6 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 113 [arrowhead = none ];
113 -> 3;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 114 [arrowhead = none ];
114 -> 4;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 115 [arrowhead = none ];
115 -> 5;

116 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 116 [arrowhead = none ];
116 -> 109;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 117 [arrowhead = none ];
117 -> 110;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 118 [arrowhead = none ];
118 -> 6;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 119 [arrowhead = none ];
119 -> 7;

120 [label="EVAL with clause\npart(X42, .(X43, X44), .(X43, X45), X46) :- ','(gt(X42, X43), part(X42, X44, X45, X46)).\nand substitution\nT6 -> T22\nX42 -> T22\nX43 -> T23\nX44 -> T24\nT7 -> .(T23, T24)\nX45 -> X47\nX7 -> .(T23, X47)\nX8 -> X48\nX46 -> X48", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 120 [arrowhead = none ];
120 -> 8;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 121 [arrowhead = none ];
121 -> 9;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 122 [arrowhead = none ];
122 -> 85;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 123 [arrowhead = none ];
123 -> 86;

124 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 124 [arrowhead = none ];
124 -> 10;

125 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
8 -> 125 [arrowhead = none ];
125 -> 11;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 126 [arrowhead = none ];
126 -> 12;

127 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 127 [arrowhead = none ];
127 -> 20;

128 [label="SPLIT 2\nnew knowledge:\nT44 is ground\nT45 is ground\nreplacements:\nX47 -> T44\nX48 -> T45", fontsize=14, style = filled, fillcolor = violet];
11 -> 128 [arrowhead = none ];
128 -> 21;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 129 [arrowhead = none ];
129 -> 13;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 130 [arrowhead = none ];
130 -> 14;

131 [label="EVAL with clause\ngt(s(X66), s(X67)) :- gt(X66, X67).\nand substitution\nX66 -> T37\nT22 -> s(T37)\nX67 -> T38\nT23 -> s(T38)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 131 [arrowhead = none ];
131 -> 15;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 132 [arrowhead = none ];
132 -> 16;

133 [label="EVAL with clause\ngt(s(0), 0).\nand substitution\nT22 -> s(0)\nT23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 133 [arrowhead = none ];
133 -> 17;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 134 [arrowhead = none ];
134 -> 18;

135 [label="INSTANCE with matching:\nT22 -> T37\nT23 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 135 [arrowhead = none , style=dashed];
135 -> 10[style=dashed];

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 136 [arrowhead = none ];
136 -> 19;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 137 [arrowhead = none ];
137 -> 22;

138 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="SPLIT 2\nnew knowledge:\nT115 is ground\nreplacements:\nX9 -> T115", fontsize=14, style = filled, fillcolor = violet];
21 -> 139 [arrowhead = none ];
139 -> 52;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 140 [arrowhead = none ];
140 -> 23;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 141 [arrowhead = none ];
141 -> 24;

142 [label="EVAL with clause\npart(X114, .(X115, X116), .(X115, X117), X118) :- ','(gt(X114, X115), part(X114, X116, X117, X118)).\nand substitution\nT22 -> T61\nX114 -> T61\nX115 -> T62\nX116 -> T63\nT24 -> .(T62, T63)\nX117 -> X119\nX47 -> .(T62, X119)\nX48 -> X120\nX118 -> X120", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 142 [arrowhead = none ];
142 -> 25;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 143 [arrowhead = none ];
143 -> 26;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 144 [arrowhead = none ];
144 -> 29;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 145 [arrowhead = none ];
145 -> 30;

146 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 146 [arrowhead = none ];
146 -> 27;

147 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
25 -> 147 [arrowhead = none ];
147 -> 28;

148 [label="INSTANCE with matching:\nT22 -> T61\nT23 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 148 [arrowhead = none , style=dashed];
148 -> 10[style=dashed];

149 [label="INSTANCE with matching:\nT22 -> T61\nT24 -> T63\nX47 -> X119\nX48 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 149 [arrowhead = none , style=dashed];
149 -> 20[style=dashed];

150 [label="EVAL with clause\npart(X165, .(X166, X167), X168, .(X166, X169)) :- ','(le(X165, X166), part(X165, X167, X168, X169)).\nand substitution\nT22 -> T83\nX165 -> T83\nX166 -> T84\nX167 -> T85\nT24 -> .(T84, T85)\nX47 -> X170\nX168 -> X170\nX169 -> X171\nX48 -> .(T84, X171)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 150 [arrowhead = none ];
150 -> 31;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 151 [arrowhead = none ];
151 -> 32;

152 [label="EVAL with clause\npart(X210, [], [], []).\nand substitution\nT22 -> T112\nX210 -> T112\nT24 -> []\nX47 -> []\nX48 -> []", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 152 [arrowhead = none ];
152 -> 48;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 153 [arrowhead = none ];
153 -> 49;

154 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 154 [arrowhead = none ];
154 -> 33;

155 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
31 -> 155 [arrowhead = none ];
155 -> 34;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 156 [arrowhead = none ];
156 -> 35;

157 [label="INSTANCE with matching:\nT22 -> T83\nT24 -> T85\nX47 -> X170\nX48 -> X171", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 157 [arrowhead = none , style=dashed];
157 -> 20[style=dashed];

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 158 [arrowhead = none ];
158 -> 36;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 159 [arrowhead = none ];
159 -> 37;

160 [label="EVAL with clause\nle(s(X189), s(X190)) :- le(X189, X190).\nand substitution\nX189 -> T98\nT83 -> s(T98)\nX190 -> T99\nT84 -> s(T99)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 160 [arrowhead = none ];
160 -> 38;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 161 [arrowhead = none ];
161 -> 39;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 162 [arrowhead = none ];
162 -> 40;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 163 [arrowhead = none ];
163 -> 41;

164 [label="INSTANCE with matching:\nT83 -> T98\nT84 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 164 [arrowhead = none , style=dashed];
164 -> 33[style=dashed];

165 [label="EVAL with clause\nle(0, s(X199)).\nand substitution\nT83 -> 0\nX199 -> T106\nT84 -> s(T106)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 165 [arrowhead = none ];
165 -> 42;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 166 [arrowhead = none ];
166 -> 43;

167 [label="EVAL with clause\nle(0, 0).\nand substitution\nT83 -> 0\nT84 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 167 [arrowhead = none ];
167 -> 45;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 168 [arrowhead = none ];
168 -> 46;

169 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 169 [arrowhead = none ];
169 -> 44;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 170 [arrowhead = none ];
170 -> 47;

171 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 171 [arrowhead = none ];
171 -> 50;

172 [label="INSTANCE with matching:\nT1 -> .(T23, T44)\nT2 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 172 [arrowhead = none , style=dashed];
172 -> 1[style=dashed];

173 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 173 [arrowhead = none ];
173 -> 53;

174 [label="SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:\nX10 -> T120", fontsize=14, style = filled, fillcolor = violet];
52 -> 174 [arrowhead = none ];
174 -> 54;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 175 [arrowhead = none ];
175 -> 55;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 176 [arrowhead = none ];
176 -> 77;

177 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 177 [arrowhead = none ];
177 -> 56;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 178 [arrowhead = none ];
178 -> 57;

179 [label="EVAL with clause\nqs(.(X268, X269), X270) :- ','(part(X268, X269, X271, X272), ','(qs(X271, X273), ','(qs(X272, X274), app(X273, .(X268, X274), X270)))).\nand substitution\nX268 -> T131\nX269 -> T132\nT45 -> .(T131, T132)\nX10 -> X275\nX270 -> X275", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 179 [arrowhead = none ];
179 -> 58;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 180 [arrowhead = none ];
180 -> 59;

181 [label="EVAL with clause\nqs([], []).\nand substitution\nT45 -> []\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 181 [arrowhead = none ];
181 -> 74;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 182 [arrowhead = none ];
182 -> 75;

183 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 183 [arrowhead = none ];
183 -> 60;

184 [label="SPLIT 2\nnew knowledge:\nT136 is ground\nT137 is ground\nreplacements:\nX271 -> T136\nX272 -> T137", fontsize=14, style = filled, fillcolor = violet];
58 -> 184 [arrowhead = none ];
184 -> 61;

185 [label="INSTANCE with matching:\nT22 -> T131\nT24 -> T132\nX47 -> X271\nX48 -> X272", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 185 [arrowhead = none , style=dashed];
185 -> 20[style=dashed];

186 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 186 [arrowhead = none ];
186 -> 62;

187 [label="SPLIT 2\nnew knowledge:\nT143 is ground\nreplacements:\nX273 -> T143", fontsize=14, style = filled, fillcolor = violet];
61 -> 187 [arrowhead = none ];
187 -> 63;

188 [label="INSTANCE with matching:\nT45 -> T136\nX10 -> X273", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 188 [arrowhead = none , style=dashed];
188 -> 53[style=dashed];

189 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
63 -> 189 [arrowhead = none ];
189 -> 64;

190 [label="SPLIT 2\nnew knowledge:\nT148 is ground\nreplacements:\nX274 -> T148", fontsize=14, style = filled, fillcolor = violet];
63 -> 190 [arrowhead = none ];
190 -> 65;

191 [label="INSTANCE with matching:\nT45 -> T137\nX10 -> X274", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 191 [arrowhead = none , style=dashed];
191 -> 53[style=dashed];

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 192 [arrowhead = none ];
192 -> 66;

193 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 193 [arrowhead = none ];
193 -> 67;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 194 [arrowhead = none ];
194 -> 68;

195 [label="EVAL with clause\napp(.(X350, X351), X352, .(X350, X353)) :- app(X351, X352, X353).\nand substitution\nX350 -> T171\nX351 -> T172\nT143 -> .(T171, T172)\nT131 -> T173\nT148 -> T174\nX352 -> .(T173, T174)\nX353 -> X354\nX275 -> .(T171, X354)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 195 [arrowhead = none ];
195 -> 69;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 196 [arrowhead = none ];
196 -> 70;

197 [label="EVAL with clause\napp([], X366, X366).\nand substitution\nT143 -> []\nT131 -> T183\nT148 -> T184\nX366 -> .(T183, T184)\nX275 -> .(T183, T184)", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 197 [arrowhead = none ];
197 -> 71;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 198 [arrowhead = none ];
198 -> 72;

199 [label="INSTANCE with matching:\nT143 -> T172\nT131 -> T173\nT148 -> T174\nX275 -> X354", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 199 [arrowhead = none , style=dashed];
199 -> 65[style=dashed];

200 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 200 [arrowhead = none ];
200 -> 73;

201 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 201 [arrowhead = none ];
201 -> 76;

202 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 202 [arrowhead = none ];
202 -> 78;

203 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 203 [arrowhead = none ];
203 -> 79;

204 [label="EVAL with clause\napp(.(X388, X389), X390, .(X388, X391)) :- app(X389, X390, X391).\nand substitution\nX388 -> T210\nX389 -> T211\nT115 -> .(T210, T211)\nT22 -> T212\nT120 -> T213\nX390 -> .(T212, T213)\nX391 -> T215\nT9 -> .(T210, T215)\nT214 -> T215", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 204 [arrowhead = none ];
204 -> 80;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 205 [arrowhead = none ];
205 -> 81;

206 [label="EVAL with clause\napp([], X402, X402).\nand substitution\nT115 -> []\nT22 -> T225\nT120 -> T226\nX402 -> .(T225, T226)\nT9 -> .(T225, T226)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 206 [arrowhead = none ];
206 -> 82;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 207 [arrowhead = none ];
207 -> 83;

208 [label="INSTANCE with matching:\nT115 -> T211\nT22 -> T212\nT120 -> T213\nT9 -> T215", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 208 [arrowhead = none , style=dashed];
208 -> 54[style=dashed];

209 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 209 [arrowhead = none ];
209 -> 84;

210 [label="EVAL with clause\npart(X432, .(X433, X434), X435, .(X433, X436)) :- ','(le(X432, X433), part(X432, X434, X435, X436)).\nand substitution\nT6 -> T239\nX432 -> T239\nX433 -> T240\nX434 -> T241\nT7 -> .(T240, T241)\nX7 -> X437\nX435 -> X437\nX436 -> X438\nX8 -> .(T240, X438)", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 210 [arrowhead = none ];
210 -> 87;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 211 [arrowhead = none ];
211 -> 88;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 212 [arrowhead = none ];
212 -> 95;

213 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 213 [arrowhead = none ];
213 -> 96;

214 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 214 [arrowhead = none ];
214 -> 89;

215 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
87 -> 215 [arrowhead = none ];
215 -> 90;

216 [label="INSTANCE with matching:\nT83 -> T239\nT84 -> T240", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 216 [arrowhead = none , style=dashed];
216 -> 33[style=dashed];

217 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
90 -> 217 [arrowhead = none ];
217 -> 91;

218 [label="SPLIT 2\nnew knowledge:\nT249 is ground\nT250 is ground\nreplacements:\nX437 -> T249\nX438 -> T250", fontsize=14, style = filled, fillcolor = violet];
90 -> 218 [arrowhead = none ];
218 -> 92;

219 [label="INSTANCE with matching:\nT22 -> T239\nT24 -> T241\nX47 -> X437\nX48 -> X438", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 219 [arrowhead = none , style=dashed];
219 -> 20[style=dashed];

220 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 220 [arrowhead = none ];
220 -> 93;

221 [label="SPLIT 2\nnew knowledge:\nT256 is ground\nreplacements:\nX9 -> T256", fontsize=14, style = filled, fillcolor = violet];
92 -> 221 [arrowhead = none ];
221 -> 94;

222 [label="INSTANCE with matching:\nT45 -> T249\nX10 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 222 [arrowhead = none , style=dashed];
222 -> 53[style=dashed];

223 [label="INSTANCE with matching:\nT45 -> .(T240, T250)\nT115 -> T256\nT22 -> T239", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 223 [arrowhead = none , style=dashed];
223 -> 52[style=dashed];

224 [label="EVAL with clause\npart(X490, [], [], []).\nand substitution\nT6 -> T265\nX490 -> T265\nT7 -> []\nX7 -> []\nX8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 224 [arrowhead = none ];
224 -> 97;

225 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 225 [arrowhead = none ];
225 -> 98;

226 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 226 [arrowhead = none ];
226 -> 107;

227 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
97 -> 227 [arrowhead = none ];
227 -> 99;

228 [label="SPLIT 2\nnew knowledge:\nT268 is ground\nreplacements:\nX9 -> T268", fontsize=14, style = filled, fillcolor = violet];
97 -> 228 [arrowhead = none ];
228 -> 100;

229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 229 [arrowhead = none ];
229 -> 101;

230 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
100 -> 230 [arrowhead = none ];
230 -> 105;

231 [label="SPLIT 2\nnew knowledge:\nT273 is ground\nreplacements:\nX10 -> T273", fontsize=14, style = filled, fillcolor = violet];
100 -> 231 [arrowhead = none ];
231 -> 106;

232 [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];
101 -> 232 [arrowhead = none ];
232 -> 102;

233 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX9 -> []", fontsize=14, style = filled, fillcolor = white];
102 -> 233 [arrowhead = none ];
233 -> 103;

234 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 234 [arrowhead = none ];
234 -> 104;

235 [label="INSTANCE with matching:\nX9 -> X10", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 235 [arrowhead = none , style=dashed];
235 -> 99[style=dashed];

236 [label="INSTANCE with matching:\nT115 -> T268\nT22 -> T265\nT120 -> T273", fontsize=14, style = filled, fillcolor = lightgrey];
106 -> 236 [arrowhead = none , style=dashed];
236 -> 54[style=dashed];

237 [label="BACKTRACK\nfor clause: qs([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
107 -> 237 [arrowhead = none ];
237 -> 108;

238 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 238 [arrowhead = none ];
238 -> 111;

}

(2) Obligation:

Clauses:

gt10(s(T37), s(T38)) :- gt10(T37, T38).
gt10(s(0), 0).
part20(T61, .(T62, T63), .(T62, X119), X120) :- gt10(T61, T62).
part20(T61, .(T62, T63), .(T62, X119), X120) :- ','(gt10(T61, T62), part20(T61, T63, X119, X120)).
part20(T83, .(T84, T85), X170, .(T84, X171)) :- le33(T83, T84).
part20(T83, .(T84, T85), X170, .(T84, X171)) :- ','(le33(T83, T84), part20(T83, T85, X170, X171)).
part20(T112, [], [], []).
le33(s(T98), s(T99)) :- le33(T98, T99).
le33(0, s(T106)).
le33(0, 0).
qs53(.(T131, T132), X275) :- part20(T131, T132, X271, X272).
qs53(.(T131, T132), X275) :- ','(part20(T131, T132, T136, T137), qs53(T136, X273)).
qs53(.(T131, T132), X275) :- ','(part20(T131, T132, T136, T137), ','(qs53(T136, T143), qs53(T137, X274))).
qs53(.(T131, T132), X275) :- ','(part20(T131, T132, T136, T137), ','(qs53(T136, T143), ','(qs53(T137, T148), app65(T143, T131, T148, X275)))).
qs53([], []).
app65(.(T171, T172), T173, T174, .(T171, X354)) :- app65(T172, T173, T174, X354).
app65([], T183, T184, .(T183, T184)).
app54(.(T210, T211), T212, T213, .(T210, T215)) :- app54(T211, T212, T213, T215).
app54([], T225, T226, .(T225, T226)).
p52(T45, X10, T115, T22, T9) :- qs53(T45, X10).
p52(T45, T120, T115, T22, T9) :- ','(qs53(T45, T120), app54(T115, T22, T120, T9)).
qs99([]).
qs1(.(T22, .(T23, T24)), T9) :- gt10(T22, T23).
qs1(.(T22, .(T23, T24)), T9) :- ','(gt10(T22, T23), part20(T22, T24, X47, X48)).
qs1(.(T22, .(T23, T24)), T9) :- ','(gt10(T22, T23), ','(part20(T22, T24, T44, T45), qs1(.(T23, T44), X9))).
qs1(.(T22, .(T23, T24)), T9) :- ','(gt10(T22, T23), ','(part20(T22, T24, T44, T45), ','(qs1(.(T23, T44), T115), p52(T45, X10, T115, T22, T9)))).
qs1(.(T239, .(T240, T241)), T9) :- le33(T239, T240).
qs1(.(T239, .(T240, T241)), T9) :- ','(le33(T239, T240), part20(T239, T241, X437, X438)).
qs1(.(T239, .(T240, T241)), T9) :- ','(le33(T239, T240), ','(part20(T239, T241, T249, T250), qs53(T249, X9))).
qs1(.(T239, .(T240, T241)), T9) :- ','(le33(T239, T240), ','(part20(T239, T241, T249, T250), ','(qs53(T249, T256), p52(.(T240, T250), X10, T256, T239, T9)))).
qs1(.(T265, []), T9) :- qs99(X9).
qs1(.(T265, []), T9) :- ','(qs99(T268), qs99(X10)).
qs1(.(T265, []), T9) :- ','(qs99(T268), ','(qs99(T273), app54(T268, T265, T273, T9))).
qs1([], []).

Queries:

qs1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (b,f) (f,f)
gt10_in: (b,b) (f,f) (b,f)
part20_in: (b,b,f,f) (b,f,f,f) (f,f,f,f)
le33_in: (b,b) (b,f) (f,f)
qs53_in: (f,f)
app65_in: (f,f,f,f)
p52_in: (f,f,f,b,f)
app54_in: (f,b,f,f) (b,f,b,f) (b,b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(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_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(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_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, gt10_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → GT10_IN_GG(T22, T23)
GT10_IN_GG(s(T37), s(T38)) → U1_GG(T37, T38, gt10_in_gg(T37, T38))
GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → PART20_IN_GGAA(T22, T24, X47, X48)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GG(T61, T62)
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_GGAA(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GG(T83, T84)
LE33_IN_GG(s(T98), s(T99)) → U6_GG(T98, T99, le33_in_gg(T98, T99))
LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_GGAA(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → GT10_IN_AA(T22, T23)
GT10_IN_AA(s(T37), s(T38)) → U1_AA(T37, T38, gt10_in_aa(T37, T38))
GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → PART20_IN_GAAA(T22, T24, X47, X48)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GA(T61, T62)
GT10_IN_GA(s(T37), s(T38)) → U1_GA(T37, T38, gt10_in_ga(T37, T38))
GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_GAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GA(T83, T84)
LE33_IN_GA(s(T98), s(T99)) → U6_GA(T98, T99, le33_in_ga(T98, T99))
LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_GAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T239, .(T240, T241)), T9) → U25_AA(T239, T240, T241, T9, le33_in_aa(T239, T240))
QS1_IN_AA(.(T239, .(T240, T241)), T9) → LE33_IN_AA(T239, T240)
LE33_IN_AA(s(T98), s(T99)) → U6_AA(T98, T99, le33_in_aa(T98, T99))
LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → PART20_IN_GAAA(T239, T241, X437, X438)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_AA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
QS53_IN_AA(.(T131, T132), X275) → U7_AA(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
QS53_IN_AA(.(T131, T132), X275) → PART20_IN_AAAA(T131, T132, X271, X272)
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_AAAA(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_AA(T61, T62)
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_AAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_AAAA(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_AA(T83, T84)
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_AAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_AA(T131, T132, X275, qs53_in_aa(T136, X273))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_AA(T131, T132, X275, qs53_in_aa(T137, X274))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_AA(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_AA(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → APP65_IN_AAAA(T143, T131, T148, X275)
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → U14_AAAA(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_AA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_AA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
P52_IN_AAAGA(T45, X10, T115, T22, T9) → U16_AAAGA(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
P52_IN_AAAGA(T45, X10, T115, T22, T9) → QS53_IN_AA(T45, X10)
P52_IN_AAAGA(T45, T120, T115, T22, T9) → U17_AAAGA(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_AAAGA(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → APP54_IN_AGAA(T115, T22, T120, T9)
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → U15_AGAA(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)
QS1_IN_AA(.(T265, []), T9) → U31_AA(T265, T9, qs99_in_a(X9))
QS1_IN_AA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_AA(.(T265, []), T9) → U32_AA(T265, T9, qs99_in_a(T268))
U32_AA(T265, T9, qs99_out_a(T268)) → U33_AA(T265, T9, qs99_in_a(X10))
U32_AA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_AA(T265, T9, qs99_out_a(T268)) → U34_AA(T265, T9, T268, qs99_in_a(T273))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → U35_AA(T265, T9, app54_in_gaga(T268, T265, T273, T9))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GAGA(T268, T265, T273, T9)
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GAGA(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_AA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_AA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_GA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_GA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
QS1_IN_GA(.(T239, .(T240, T241)), T9) → U25_GA(T239, T240, T241, T9, le33_in_gg(T239, T240))
QS1_IN_GA(.(T239, .(T240, T241)), T9) → LE33_IN_GG(T239, T240)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → PART20_IN_GGAA(T239, T241, X437, X438)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_GA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_GA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_GA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
QS1_IN_GA(.(T265, []), T9) → U31_GA(T265, T9, qs99_in_a(X9))
QS1_IN_GA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_GA(.(T265, []), T9) → U32_GA(T265, T9, qs99_in_a(T268))
U32_GA(T265, T9, qs99_out_a(T268)) → U33_GA(T265, T9, qs99_in_a(X10))
U32_GA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_GA(T265, T9, qs99_out_a(T268)) → U34_GA(T265, T9, T268, qs99_in_a(T273))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → U35_GA(T265, T9, app54_in_ggga(T268, T265, T273, T9))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GGGA(T268, T265, T273, T9)
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GGGA(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x2, x3, x5)
GT10_IN_GG(x1, x2)  =  GT10_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
LE33_IN_GG(x1, x2)  =  LE33_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x2, x3, x5)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(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)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x1, x5)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
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)
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4)  =  U13_AA(x4)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA
U14_AAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAA(x6)
U29_AA(x1, x2, x3, x4, x5, x6)  =  U29_AA(x1, x6)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
P52_IN_AAAGA(x1, x2, x3, x4, x5)  =  P52_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x4, x6)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)
U15_AGAA(x1, x2, x3, x4, x5, x6)  =  U15_AGAA(x3, x6)
U31_AA(x1, x2, x3)  =  U31_AA(x3)
QS99_IN_A(x1)  =  QS99_IN_A
U32_AA(x1, x2, x3)  =  U32_AA(x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x3, x4)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)
U15_GAGA(x1, x2, x3, x4, x5, x6)  =  U15_GAGA(x1, x2, x4, x6)
U23_AA(x1, x2, x3, x4, x5, x6)  =  U23_AA(x1, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U23_GA(x1, x2, x3, x4, x5, x6)  =  U23_GA(x1, x2, x3, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_GA(x1, x2, x3, x5)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x2, x3, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x1, x3, x4)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)
U15_GGGA(x1, x2, x3, x4, x5, x6)  =  U15_GGGA(x1, x2, x3, x4, x6)

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

(6) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, gt10_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → GT10_IN_GG(T22, T23)
GT10_IN_GG(s(T37), s(T38)) → U1_GG(T37, T38, gt10_in_gg(T37, T38))
GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → PART20_IN_GGAA(T22, T24, X47, X48)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GG(T61, T62)
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_GGAA(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GG(T83, T84)
LE33_IN_GG(s(T98), s(T99)) → U6_GG(T98, T99, le33_in_gg(T98, T99))
LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_GGAA(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → GT10_IN_AA(T22, T23)
GT10_IN_AA(s(T37), s(T38)) → U1_AA(T37, T38, gt10_in_aa(T37, T38))
GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → PART20_IN_GAAA(T22, T24, X47, X48)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GA(T61, T62)
GT10_IN_GA(s(T37), s(T38)) → U1_GA(T37, T38, gt10_in_ga(T37, T38))
GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_GAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GA(T83, T84)
LE33_IN_GA(s(T98), s(T99)) → U6_GA(T98, T99, le33_in_ga(T98, T99))
LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_GAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T239, .(T240, T241)), T9) → U25_AA(T239, T240, T241, T9, le33_in_aa(T239, T240))
QS1_IN_AA(.(T239, .(T240, T241)), T9) → LE33_IN_AA(T239, T240)
LE33_IN_AA(s(T98), s(T99)) → U6_AA(T98, T99, le33_in_aa(T98, T99))
LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → PART20_IN_GAAA(T239, T241, X437, X438)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_AA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
QS53_IN_AA(.(T131, T132), X275) → U7_AA(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
QS53_IN_AA(.(T131, T132), X275) → PART20_IN_AAAA(T131, T132, X271, X272)
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_AAAA(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_AA(T61, T62)
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_AAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_AAAA(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_AA(T83, T84)
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_AAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_AA(T131, T132, X275, qs53_in_aa(T136, X273))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_AA(T131, T132, X275, qs53_in_aa(T137, X274))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_AA(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_AA(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → APP65_IN_AAAA(T143, T131, T148, X275)
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → U14_AAAA(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_AA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_AA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
P52_IN_AAAGA(T45, X10, T115, T22, T9) → U16_AAAGA(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
P52_IN_AAAGA(T45, X10, T115, T22, T9) → QS53_IN_AA(T45, X10)
P52_IN_AAAGA(T45, T120, T115, T22, T9) → U17_AAAGA(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_AAAGA(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → APP54_IN_AGAA(T115, T22, T120, T9)
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → U15_AGAA(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)
QS1_IN_AA(.(T265, []), T9) → U31_AA(T265, T9, qs99_in_a(X9))
QS1_IN_AA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_AA(.(T265, []), T9) → U32_AA(T265, T9, qs99_in_a(T268))
U32_AA(T265, T9, qs99_out_a(T268)) → U33_AA(T265, T9, qs99_in_a(X10))
U32_AA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_AA(T265, T9, qs99_out_a(T268)) → U34_AA(T265, T9, T268, qs99_in_a(T273))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → U35_AA(T265, T9, app54_in_gaga(T268, T265, T273, T9))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GAGA(T268, T265, T273, T9)
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GAGA(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_AA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_AA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_GA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_GA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
QS1_IN_GA(.(T239, .(T240, T241)), T9) → U25_GA(T239, T240, T241, T9, le33_in_gg(T239, T240))
QS1_IN_GA(.(T239, .(T240, T241)), T9) → LE33_IN_GG(T239, T240)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → PART20_IN_GGAA(T239, T241, X437, X438)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_GA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_GA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_GA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
QS1_IN_GA(.(T265, []), T9) → U31_GA(T265, T9, qs99_in_a(X9))
QS1_IN_GA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_GA(.(T265, []), T9) → U32_GA(T265, T9, qs99_in_a(T268))
U32_GA(T265, T9, qs99_out_a(T268)) → U33_GA(T265, T9, qs99_in_a(X10))
U32_GA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_GA(T265, T9, qs99_out_a(T268)) → U34_GA(T265, T9, T268, qs99_in_a(T273))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → U35_GA(T265, T9, app54_in_ggga(T268, T265, T273, T9))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GGGA(T268, T265, T273, T9)
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GGGA(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x2, x3, x5)
GT10_IN_GG(x1, x2)  =  GT10_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
LE33_IN_GG(x1, x2)  =  LE33_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x2, x3, x5)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(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)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x1, x5)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
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)
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4)  =  U13_AA(x4)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA
U14_AAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAA(x6)
U29_AA(x1, x2, x3, x4, x5, x6)  =  U29_AA(x1, x6)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
P52_IN_AAAGA(x1, x2, x3, x4, x5)  =  P52_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x4, x6)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)
U15_AGAA(x1, x2, x3, x4, x5, x6)  =  U15_AGAA(x3, x6)
U31_AA(x1, x2, x3)  =  U31_AA(x3)
QS99_IN_A(x1)  =  QS99_IN_A
U32_AA(x1, x2, x3)  =  U32_AA(x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x3, x4)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)
U15_GAGA(x1, x2, x3, x4, x5, x6)  =  U15_GAGA(x1, x2, x4, x6)
U23_AA(x1, x2, x3, x4, x5, x6)  =  U23_AA(x1, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U23_GA(x1, x2, x3, x4, x5, x6)  =  U23_GA(x1, x2, x3, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_GA(x1, x2, x3, x5)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x2, x3, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x1, x3, x4)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)
U15_GGGA(x1, x2, x3, x4, x5, x6)  =  U15_GGGA(x1, x2, x3, x4, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 91 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP54_IN_GGGA(.(T210, T211), T212, T213) → APP54_IN_GGGA(T211, T212, T213)

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:

• APP54_IN_GGGA(.(T210, T211), T212, T213) → APP54_IN_GGGA(T211, T212, T213)
  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:

APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)

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:

APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)

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:

APP54_IN_GAGA(.(T210, T211), T213) → APP54_IN_GAGA(T211, T213)

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:

• APP54_IN_GAGA(.(T210, T211), T213) → APP54_IN_GAGA(T211, T213)
  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:

APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)

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:

APP54_IN_AGAA(T212) → APP54_IN_AGAA(T212)

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 = APP54_IN_AGAA(T212) evaluates to t =APP54_IN_AGAA(T212)

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 APP54_IN_AGAA(T212) to APP54_IN_AGAA(T212).

(29) NO

(30) Obligation:

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

APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP65_IN_AAAA → APP65_IN_AAAA

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

(35) NonTerminationProof (EQUIVALENT transformation)

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

s = APP65_IN_AAAA evaluates to t =APP65_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 APP65_IN_AAAA to APP65_IN_AAAA.

(36) NO

(37) Obligation:

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

LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA

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:

LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)

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

(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:

LE33_IN_AA → LE33_IN_AA

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 = 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.

(43) NO

(44) Obligation:

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

LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)

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

(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:

LE33_IN_GA(s(T98)) → LE33_IN_GA(T98)

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

(49) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LE33_IN_GA(s(T98)) → LE33_IN_GA(T98)
  The graph contains the following edges 1 > 1

(50) YES

(51) Obligation:

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

GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)

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:

GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)

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

(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:

GT10_IN_GA(s(T37)) → GT10_IN_GA(T37)

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

(56) 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(T37)) → GT10_IN_GA(T37)
  The graph contains the following edges 1 > 1

(57) YES

(58) Obligation:

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

U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)

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:

U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
gt10_out_ga(x1, x2)  =  gt10_out_ga(x1, x2)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)

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:

U2_GAAA(T61, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61)
PART20_IN_GAAA(T61) → U2_GAAA(T61, gt10_in_ga(T61))
PART20_IN_GAAA(T83) → U4_GAAA(T83, le33_in_ga(T83))
U4_GAAA(T83, le33_out_ga(T83)) → PART20_IN_GAAA(T83)

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(63) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T61) → U2_GAAA(T61, gt10_in_ga(T61)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))

(64) Obligation:

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

U2_GAAA(T61, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61)
PART20_IN_GAAA(T83) → U4_GAAA(T83, le33_in_ga(T83))
U4_GAAA(T83, le33_out_ga(T83)) → PART20_IN_GAAA(T83)
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(65) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T83) → U4_GAAA(T83, le33_in_ga(T83)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0))

(66) Obligation:

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

U2_GAAA(T61, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61)
U4_GAAA(T83, le33_out_ga(T83)) → PART20_IN_GAAA(T83)
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(67) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GAAA(T61, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61) we obtained the following new rules [LPAR04]:

U2_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U2_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))

(68) Obligation:

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

U4_GAAA(T83, le33_out_ga(T83)) → PART20_IN_GAAA(T83)
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0))
U2_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U2_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(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(69) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GAAA(T83, le33_out_ga(T83)) → PART20_IN_GAAA(T83) we obtained the following new rules [LPAR04]:

U4_GAAA(s(z0), le33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))
U4_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)

(70) Obligation:

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

PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(x0, le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0))
U2_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U2_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
U4_GAAA(s(z0), le33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))
U4_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(71) DependencyGraphProof (EQUIVALENT transformation)

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

(72) Complex Obligation (AND)

(73) Obligation:

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

U4_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(74) 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.

(75) Obligation:

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

U4_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U4_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)
U6_ga(x0, x1)

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

(76) 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)
U6_ga(x0, x1)

(77) Obligation:

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

U4_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0))

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

(78) 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) → U4_GAAA(0, le33_out_ga(0))
with rule PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga(0)) at position [] and matcher [ ]

U4_GAAA(0, le33_out_ga(0)) → PART20_IN_GAAA(0)
with rule U4_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.

(79) NO

(80) Obligation:

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

U2_GAAA(s(z0), gt10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(x0, gt10_in_ga(x0)))
U2_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(x0, le33_in_ga(x0)))
U4_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(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

gt10_in_ga(x0)
le33_in_ga(x0)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(81) NonTerminationProof (EQUIVALENT transformation)

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0))
with rule PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(s(0), 0)) at position [] and matcher [ ]

U2_GAAA(s(0), gt10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
with rule U2_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.

(82) NO

(83) Obligation:

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

GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA

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

(84) UsableRulesProof (EQUIVALENT transformation)

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

(85) Obligation:

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

GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)

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

(86) PiDPToQDPProof (SOUND transformation)

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

(87) Obligation:

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

GT10_IN_AA → GT10_IN_AA

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

(88) 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.

(89) NO

(90) Obligation:

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

QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)

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

(91) UsableRulesProof (EQUIVALENT transformation)

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

(92) Obligation:

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

QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)

The TRS R consists of the following rules:

part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
qs53_in_aa([], []) → qs53_out_aa([], [])
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
[]  =  []
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)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)

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

(93) PiDPToQDPProof (SOUND transformation)

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

(94) Obligation:

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

QS53_IN_AA → U8_AA(part20_in_aaaa)
U8_AA(part20_out_aaaa) → QS53_IN_AA
U8_AA(part20_out_aaaa) → U10_AA(qs53_in_aa)
U10_AA(qs53_out_aa) → QS53_IN_AA

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U4_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs53_in_aa → U7_aa(part20_in_aaaa)
qs53_in_aa → U8_aa(part20_in_aaaa)
qs53_in_aa → qs53_out_aa
U2_aaaa(gt10_out_aa(T61, T62)) → part20_out_aaaa
U2_aaaa(gt10_out_aa(T61, T62)) → U3_aaaa(part20_in_gaaa(T61))
U4_aaaa(le33_out_aa(T83)) → part20_out_aaaa
U4_aaaa(le33_out_aa(T83)) → U5_aaaa(part20_in_gaaa(T83))
U7_aa(part20_out_aaaa) → qs53_out_aa
U8_aa(part20_out_aaaa) → U9_aa(qs53_in_aa)
U8_aa(part20_out_aaaa) → U10_aa(qs53_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U3_aaaa(part20_out_gaaa(T61)) → part20_out_aaaa
le33_in_aa → U6_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U5_aaaa(part20_out_gaaa(T83)) → part20_out_aaaa
U9_aa(qs53_out_aa) → qs53_out_aa
U10_aa(qs53_out_aa) → U11_aa(qs53_in_aa)
U10_aa(qs53_out_aa) → U12_aa(qs53_in_aa)
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa(T112)
U6_aa(le33_out_aa(T98)) → le33_out_aa(s(T98))
U11_aa(qs53_out_aa) → qs53_out_aa
U12_aa(qs53_out_aa) → U13_aa(app65_in_aaaa)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → U3_gaaa(T61, part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga(T83)) → part20_out_gaaa(T83)
U4_gaaa(T83, le33_out_ga(T83)) → U5_gaaa(T83, part20_in_gaaa(T83))
U13_aa(app65_out_aaaa) → qs53_out_aa
gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, part20_out_gaaa(T61)) → part20_out_gaaa(T61)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U5_gaaa(T83, part20_out_gaaa(T83)) → part20_out_gaaa(T83)
app65_in_aaaa → U14_aaaa(app65_in_aaaa)
app65_in_aaaa → app65_out_aaaa
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))
U14_aaaa(app65_out_aaaa) → app65_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs53_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
gt10_in_aa
U3_aaaa(x0)
le33_in_aa
U5_aaaa(x0)
U9_aa(x0)
U10_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U6_aa(x0)
U11_aa(x0)
U12_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U13_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0, x1)
le33_in_ga(x0)
U5_gaaa(x0, x1)
app65_in_aaaa
U1_ga(x0, x1)
U6_ga(x0, x1)
U14_aaaa(x0)

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

(95) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS53_IN_AA → U8_AA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS53_IN_AA → U8_AA(U2_aaaa(gt10_in_aa))
QS53_IN_AA → U8_AA(U4_aaaa(le33_in_aa))
QS53_IN_AA → U8_AA(part20_out_aaaa)

(96) Obligation:

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

U8_AA(part20_out_aaaa) → QS53_IN_AA
U8_AA(part20_out_aaaa) → U10_AA(qs53_in_aa)
U10_AA(qs53_out_aa) → QS53_IN_AA
QS53_IN_AA → U8_AA(U2_aaaa(gt10_in_aa))
QS53_IN_AA → U8_AA(U4_aaaa(le33_in_aa))
QS53_IN_AA → U8_AA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U4_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs53_in_aa → U7_aa(part20_in_aaaa)
qs53_in_aa → U8_aa(part20_in_aaaa)
qs53_in_aa → qs53_out_aa
U2_aaaa(gt10_out_aa(T61, T62)) → part20_out_aaaa
U2_aaaa(gt10_out_aa(T61, T62)) → U3_aaaa(part20_in_gaaa(T61))
U4_aaaa(le33_out_aa(T83)) → part20_out_aaaa
U4_aaaa(le33_out_aa(T83)) → U5_aaaa(part20_in_gaaa(T83))
U7_aa(part20_out_aaaa) → qs53_out_aa
U8_aa(part20_out_aaaa) → U9_aa(qs53_in_aa)
U8_aa(part20_out_aaaa) → U10_aa(qs53_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U3_aaaa(part20_out_gaaa(T61)) → part20_out_aaaa
le33_in_aa → U6_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U5_aaaa(part20_out_gaaa(T83)) → part20_out_aaaa
U9_aa(qs53_out_aa) → qs53_out_aa
U10_aa(qs53_out_aa) → U11_aa(qs53_in_aa)
U10_aa(qs53_out_aa) → U12_aa(qs53_in_aa)
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa(T112)
U6_aa(le33_out_aa(T98)) → le33_out_aa(s(T98))
U11_aa(qs53_out_aa) → qs53_out_aa
U12_aa(qs53_out_aa) → U13_aa(app65_in_aaaa)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → U3_gaaa(T61, part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga(T83)) → part20_out_gaaa(T83)
U4_gaaa(T83, le33_out_ga(T83)) → U5_gaaa(T83, part20_in_gaaa(T83))
U13_aa(app65_out_aaaa) → qs53_out_aa
gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, part20_out_gaaa(T61)) → part20_out_gaaa(T61)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U5_gaaa(T83, part20_out_gaaa(T83)) → part20_out_gaaa(T83)
app65_in_aaaa → U14_aaaa(app65_in_aaaa)
app65_in_aaaa → app65_out_aaaa
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))
U14_aaaa(app65_out_aaaa) → app65_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs53_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
gt10_in_aa
U3_aaaa(x0)
le33_in_aa
U5_aaaa(x0)
U9_aa(x0)
U10_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U6_aa(x0)
U11_aa(x0)
U12_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U13_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0, x1)
le33_in_ga(x0)
U5_gaaa(x0, x1)
app65_in_aaaa
U1_ga(x0, x1)
U6_ga(x0, x1)
U14_aaaa(x0)

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

(97) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U8_AA(part20_out_aaaa) → U10_AA(qs53_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U8_AA(part20_out_aaaa) → U10_AA(U7_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(U8_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(qs53_out_aa)

(98) Obligation:

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

U8_AA(part20_out_aaaa) → QS53_IN_AA
U10_AA(qs53_out_aa) → QS53_IN_AA
QS53_IN_AA → U8_AA(U2_aaaa(gt10_in_aa))
QS53_IN_AA → U8_AA(U4_aaaa(le33_in_aa))
QS53_IN_AA → U8_AA(part20_out_aaaa)
U8_AA(part20_out_aaaa) → U10_AA(U7_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(U8_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(qs53_out_aa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U4_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs53_in_aa → U7_aa(part20_in_aaaa)
qs53_in_aa → U8_aa(part20_in_aaaa)
qs53_in_aa → qs53_out_aa
U2_aaaa(gt10_out_aa(T61, T62)) → part20_out_aaaa
U2_aaaa(gt10_out_aa(T61, T62)) → U3_aaaa(part20_in_gaaa(T61))
U4_aaaa(le33_out_aa(T83)) → part20_out_aaaa
U4_aaaa(le33_out_aa(T83)) → U5_aaaa(part20_in_gaaa(T83))
U7_aa(part20_out_aaaa) → qs53_out_aa
U8_aa(part20_out_aaaa) → U9_aa(qs53_in_aa)
U8_aa(part20_out_aaaa) → U10_aa(qs53_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U3_aaaa(part20_out_gaaa(T61)) → part20_out_aaaa
le33_in_aa → U6_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U5_aaaa(part20_out_gaaa(T83)) → part20_out_aaaa
U9_aa(qs53_out_aa) → qs53_out_aa
U10_aa(qs53_out_aa) → U11_aa(qs53_in_aa)
U10_aa(qs53_out_aa) → U12_aa(qs53_in_aa)
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa(T112)
U6_aa(le33_out_aa(T98)) → le33_out_aa(s(T98))
U11_aa(qs53_out_aa) → qs53_out_aa
U12_aa(qs53_out_aa) → U13_aa(app65_in_aaaa)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → U3_gaaa(T61, part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga(T83)) → part20_out_gaaa(T83)
U4_gaaa(T83, le33_out_ga(T83)) → U5_gaaa(T83, part20_in_gaaa(T83))
U13_aa(app65_out_aaaa) → qs53_out_aa
gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, part20_out_gaaa(T61)) → part20_out_gaaa(T61)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U5_gaaa(T83, part20_out_gaaa(T83)) → part20_out_gaaa(T83)
app65_in_aaaa → U14_aaaa(app65_in_aaaa)
app65_in_aaaa → app65_out_aaaa
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))
U14_aaaa(app65_out_aaaa) → app65_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs53_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
gt10_in_aa
U3_aaaa(x0)
le33_in_aa
U5_aaaa(x0)
U9_aa(x0)
U10_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U6_aa(x0)
U11_aa(x0)
U12_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U13_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0, x1)
le33_in_ga(x0)
U5_gaaa(x0, x1)
app65_in_aaaa
U1_ga(x0, x1)
U6_ga(x0, x1)
U14_aaaa(x0)

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

(99) 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 = QS53_IN_AA evaluates to t =QS53_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS53_IN_AA → U8_AA(part20_out_aaaa)
with rule QS53_IN_AA → U8_AA(part20_out_aaaa) at position [] and matcher [ ]

U8_AA(part20_out_aaaa) → QS53_IN_AA
with rule U8_AA(part20_out_aaaa) → QS53_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.

(100) NO

(101) Obligation:

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

U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)

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

(102) UsableRulesProof (EQUIVALENT transformation)

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

(103) Obligation:

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

U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))

The TRS R consists of the following rules:

part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
[]  =  []
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)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)

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

(104) PiDPToQDPProof (SOUND transformation)

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

(105) Obligation:

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

U19_AA(gt10_out_aa(T22, T23)) → U21_AA(T22, part20_in_gaaa(T22))
U21_AA(T22, part20_out_gaaa(T22)) → QS1_IN_AA
QS1_IN_AA → U19_AA(gt10_in_aa)

The TRS R consists of the following rules:

part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa(T112)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → U3_gaaa(T61, part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga(T83)) → part20_out_gaaa(T83)
U4_gaaa(T83, le33_out_ga(T83)) → U5_gaaa(T83, part20_in_gaaa(T83))
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, part20_out_gaaa(T61)) → part20_out_gaaa(T61)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U5_gaaa(T83, part20_out_gaaa(T83)) → part20_out_gaaa(T83)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

part20_in_gaaa(x0)
gt10_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0, x1)
le33_in_ga(x0)
U5_gaaa(x0, x1)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(106) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U19_AA(gt10_out_aa(T22, T23)) → U21_AA(T22, part20_in_gaaa(T22)) at position [1] we obtained the following new rules [LPAR04]:

U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U2_gaaa(x0, gt10_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U4_gaaa(x0, le33_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa(x0))

(107) Obligation:

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

U21_AA(T22, part20_out_gaaa(T22)) → QS1_IN_AA
QS1_IN_AA → U19_AA(gt10_in_aa)
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U2_gaaa(x0, gt10_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U4_gaaa(x0, le33_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa(x0))

The TRS R consists of the following rules:

part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa(T112)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → U3_gaaa(T61, part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga(T83)) → part20_out_gaaa(T83)
U4_gaaa(T83, le33_out_ga(T83)) → U5_gaaa(T83, part20_in_gaaa(T83))
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, part20_out_gaaa(T61)) → part20_out_gaaa(T61)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U5_gaaa(T83, part20_out_gaaa(T83)) → part20_out_gaaa(T83)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

part20_in_gaaa(x0)
gt10_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0, x1)
le33_in_ga(x0)
U5_gaaa(x0, x1)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(108) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS1_IN_AA → U19_AA(gt10_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QS1_IN_AA → U19_AA(U1_aa(gt10_in_aa))
QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0))

(109) Obligation:

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

U21_AA(T22, part20_out_gaaa(T22)) → QS1_IN_AA
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U2_gaaa(x0, gt10_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U4_gaaa(x0, le33_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa(x0))
QS1_IN_AA → U19_AA(U1_aa(gt10_in_aa))
QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0))

The TRS R consists of the following rules:

part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa(T112)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61)
U2_gaaa(T61, gt10_out_ga(T61, T62)) → U3_gaaa(T61, part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga(T83)) → part20_out_gaaa(T83)
U4_gaaa(T83, le33_out_ga(T83)) → U5_gaaa(T83, part20_in_gaaa(T83))
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37)) → U1_ga(T37, gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, part20_out_gaaa(T61)) → part20_out_gaaa(T61)
le33_in_ga(s(T98)) → U6_ga(T98, le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga(0)
U5_gaaa(T83, part20_out_gaaa(T83)) → part20_out_gaaa(T83)
U1_ga(T37, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, le33_out_ga(T98)) → le33_out_ga(s(T98))

The set Q consists of the following terms:

part20_in_gaaa(x0)
gt10_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0, x1)
le33_in_ga(x0)
U5_gaaa(x0, x1)
U1_ga(x0, x1)
U6_ga(x0, x1)

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

(110) NonTerminationProof (EQUIVALENT transformation)

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

s = QS1_IN_AA evaluates to t =QS1_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0))
with rule QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0)) at position [] and matcher [ ]

U19_AA(gt10_out_aa(s(0), 0)) → U21_AA(s(0), part20_out_gaaa(s(0)))
with rule U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa(x0)) at position [] and matcher [x0 / s(0), y1 / 0]

U21_AA(s(0), part20_out_gaaa(s(0))) → QS1_IN_AA
with rule U21_AA(T22, part20_out_gaaa(T22)) → QS1_IN_AA

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


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

(111) NO

(112) Obligation:

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

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
LE33_IN_GG(x1, x2)  =  LE33_IN_GG(x1, x2)

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

(113) UsableRulesProof (EQUIVALENT transformation)

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

(114) Obligation:

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

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

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

(115) PiDPToQDPProof (EQUIVALENT transformation)

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

(116) Obligation:

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

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

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

(117) 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_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
  The graph contains the following edges 1 > 1, 2 > 2

(118) YES

(119) Obligation:

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

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
GT10_IN_GG(x1, x2)  =  GT10_IN_GG(x1, x2)

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:

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

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

(122) PiDPToQDPProof (EQUIVALENT 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:

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

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

(124) 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_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
  The graph contains the following edges 1 > 1, 2 > 2

(125) YES

(126) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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)
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)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
le33_out_ga(x1, x2)  =  le33_out_ga(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x3, x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa(x2)
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x1, x2, x4, x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga(x1, x3)
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x2, x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x2, x3, x4, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x1, x2, x3, x4)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)

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

(127) UsableRulesProof (EQUIVALENT transformation)

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

(128) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg(x1, x2)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
le33_out_gg(x1, x2)  =  le33_out_gg(x1, x2)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)

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

(129) PiDPToQDPProof (SOUND transformation)

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

(130) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63)) → U2_GGAA(T61, T62, T63, gt10_in_gg(T61, T62))
U2_GGAA(T61, T62, T63, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63)
PART20_IN_GGAA(T83, .(T84, T85)) → U4_GGAA(T83, T84, T85, le33_in_gg(T83, T84))
U4_GGAA(T83, T84, T85, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85)

The TRS R consists of the following rules:

gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gt10_in_gg(x0, x1)
le33_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U6_gg(x0, x1, x2)

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

(131) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• U2_GGAA(T61, T62, T63, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• U4_GGAA(T83, T84, T85, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• PART20_IN_GGAA(T61, .(T62, T63)) → U2_GGAA(T61, T62, T63, gt10_in_gg(T61, T62))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

• PART20_IN_GGAA(T83, .(T84, T85)) → U4_GGAA(T83, T84, T85, le33_in_gg(T83, T84))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(132) YES

(133) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (b,f) (f,f)
gt10_in: (b,b) (f,f) (b,f)
part20_in: (b,b,f,f) (b,f,f,f) (f,f,f,f)
le33_in: (b,b) (b,f) (f,f)
qs53_in: (f,f)
app65_in: (f,f,f,f)
p52_in: (f,f,f,b,f)
app54_in: (f,b,f,f) (b,f,b,f) (b,b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(134) Obligation:

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

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)

(135) DependencyPairsProof (EQUIVALENT transformation)

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, gt10_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → GT10_IN_GG(T22, T23)
GT10_IN_GG(s(T37), s(T38)) → U1_GG(T37, T38, gt10_in_gg(T37, T38))
GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → PART20_IN_GGAA(T22, T24, X47, X48)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GG(T61, T62)
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_GGAA(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GG(T83, T84)
LE33_IN_GG(s(T98), s(T99)) → U6_GG(T98, T99, le33_in_gg(T98, T99))
LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_GGAA(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → GT10_IN_AA(T22, T23)
GT10_IN_AA(s(T37), s(T38)) → U1_AA(T37, T38, gt10_in_aa(T37, T38))
GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → PART20_IN_GAAA(T22, T24, X47, X48)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GA(T61, T62)
GT10_IN_GA(s(T37), s(T38)) → U1_GA(T37, T38, gt10_in_ga(T37, T38))
GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_GAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GA(T83, T84)
LE33_IN_GA(s(T98), s(T99)) → U6_GA(T98, T99, le33_in_ga(T98, T99))
LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_GAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T239, .(T240, T241)), T9) → U25_AA(T239, T240, T241, T9, le33_in_aa(T239, T240))
QS1_IN_AA(.(T239, .(T240, T241)), T9) → LE33_IN_AA(T239, T240)
LE33_IN_AA(s(T98), s(T99)) → U6_AA(T98, T99, le33_in_aa(T98, T99))
LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → PART20_IN_GAAA(T239, T241, X437, X438)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_AA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
QS53_IN_AA(.(T131, T132), X275) → U7_AA(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
QS53_IN_AA(.(T131, T132), X275) → PART20_IN_AAAA(T131, T132, X271, X272)
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_AAAA(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_AA(T61, T62)
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_AAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_AAAA(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_AA(T83, T84)
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_AAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_AA(T131, T132, X275, qs53_in_aa(T136, X273))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_AA(T131, T132, X275, qs53_in_aa(T137, X274))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_AA(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_AA(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → APP65_IN_AAAA(T143, T131, T148, X275)
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → U14_AAAA(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_AA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_AA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
P52_IN_AAAGA(T45, X10, T115, T22, T9) → U16_AAAGA(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
P52_IN_AAAGA(T45, X10, T115, T22, T9) → QS53_IN_AA(T45, X10)
P52_IN_AAAGA(T45, T120, T115, T22, T9) → U17_AAAGA(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_AAAGA(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → APP54_IN_AGAA(T115, T22, T120, T9)
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → U15_AGAA(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)
QS1_IN_AA(.(T265, []), T9) → U31_AA(T265, T9, qs99_in_a(X9))
QS1_IN_AA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_AA(.(T265, []), T9) → U32_AA(T265, T9, qs99_in_a(T268))
U32_AA(T265, T9, qs99_out_a(T268)) → U33_AA(T265, T9, qs99_in_a(X10))
U32_AA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_AA(T265, T9, qs99_out_a(T268)) → U34_AA(T265, T9, T268, qs99_in_a(T273))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → U35_AA(T265, T9, app54_in_gaga(T268, T265, T273, T9))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GAGA(T268, T265, T273, T9)
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GAGA(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_AA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_AA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_GA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_GA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
QS1_IN_GA(.(T239, .(T240, T241)), T9) → U25_GA(T239, T240, T241, T9, le33_in_gg(T239, T240))
QS1_IN_GA(.(T239, .(T240, T241)), T9) → LE33_IN_GG(T239, T240)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → PART20_IN_GGAA(T239, T241, X437, X438)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_GA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_GA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_GA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
QS1_IN_GA(.(T265, []), T9) → U31_GA(T265, T9, qs99_in_a(X9))
QS1_IN_GA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_GA(.(T265, []), T9) → U32_GA(T265, T9, qs99_in_a(T268))
U32_GA(T265, T9, qs99_out_a(T268)) → U33_GA(T265, T9, qs99_in_a(X10))
U32_GA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_GA(T265, T9, qs99_out_a(T268)) → U34_GA(T265, T9, T268, qs99_in_a(T273))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → U35_GA(T265, T9, app54_in_ggga(T268, T265, T273, T9))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GGGA(T268, T265, T273, T9)
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GGGA(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x3, x5)
GT10_IN_GG(x1, x2)  =  GT10_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x3, x6)
LE33_IN_GG(x1, x2)  =  LE33_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x6)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x5)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(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(x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x6)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x1, x5)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
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)
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4)  =  U13_AA(x4)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA
U14_AAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAA(x6)
U29_AA(x1, x2, x3, x4, x5, x6)  =  U29_AA(x1, x6)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
P52_IN_AAAGA(x1, x2, x3, x4, x5)  =  P52_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x6)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)
U15_AGAA(x1, x2, x3, x4, x5, x6)  =  U15_AGAA(x6)
U31_AA(x1, x2, x3)  =  U31_AA(x3)
QS99_IN_A(x1)  =  QS99_IN_A
U32_AA(x1, x2, x3)  =  U32_AA(x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x3, x4)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)
U15_GAGA(x1, x2, x3, x4, x5, x6)  =  U15_GAGA(x6)
U23_AA(x1, x2, x3, x4, x5, x6)  =  U23_AA(x1, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U23_GA(x1, x2, x3, x4, x5, x6)  =  U23_GA(x1, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x3, x5)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_GA(x5)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x5)
U31_GA(x1, x2, x3)  =  U31_GA(x3)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x3)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x1, x3, x4)
U35_GA(x1, x2, x3)  =  U35_GA(x3)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)
U15_GGGA(x1, x2, x3, x4, x5, x6)  =  U15_GGGA(x1, x6)

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

(136) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, gt10_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → GT10_IN_GG(T22, T23)
GT10_IN_GG(s(T37), s(T38)) → U1_GG(T37, T38, gt10_in_gg(T37, T38))
GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → PART20_IN_GGAA(T22, T24, X47, X48)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GG(T61, T62)
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_GGAA(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GG(T83, T84)
LE33_IN_GG(s(T98), s(T99)) → U6_GG(T98, T99, le33_in_gg(T98, T99))
LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_GGAA(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)
U19_GA(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_GA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))
QS1_IN_AA(.(T22, .(T23, T24)), T9) → GT10_IN_AA(T22, T23)
GT10_IN_AA(s(T37), s(T38)) → U1_AA(T37, T38, gt10_in_aa(T37, T38))
GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → PART20_IN_GAAA(T22, T24, X47, X48)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GA(T61, T62)
GT10_IN_GA(s(T37), s(T38)) → U1_GA(T37, T38, gt10_in_ga(T37, T38))
GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_GAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GA(T83, T84)
LE33_IN_GA(s(T98), s(T99)) → U6_GA(T98, T99, le33_in_ga(T98, T99))
LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_GAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_AA(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T239, .(T240, T241)), T9) → U25_AA(T239, T240, T241, T9, le33_in_aa(T239, T240))
QS1_IN_AA(.(T239, .(T240, T241)), T9) → LE33_IN_AA(T239, T240)
LE33_IN_AA(s(T98), s(T99)) → U6_AA(T98, T99, le33_in_aa(T98, T99))
LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → PART20_IN_GAAA(T239, T241, X437, X438)
U25_AA(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_AA(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_AA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
QS53_IN_AA(.(T131, T132), X275) → U7_AA(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
QS53_IN_AA(.(T131, T132), X275) → PART20_IN_AAAA(T131, T132, X271, X272)
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_AAAA(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
PART20_IN_AAAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_AA(T61, T62)
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_AAAA(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U2_AAAA(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_AAAA(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
PART20_IN_AAAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_AA(T83, T84)
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_AAAA(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U4_AAAA(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)
QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_AA(T131, T132, X275, qs53_in_aa(T136, X273))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_AA(T131, T132, X275, qs53_in_aa(T137, X274))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_AA(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_AA(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
U12_AA(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → APP65_IN_AAAA(T143, T131, T148, X275)
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → U14_AAAA(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)
U27_AA(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_AA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_AA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_AA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
P52_IN_AAAGA(T45, X10, T115, T22, T9) → U16_AAAGA(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
P52_IN_AAAGA(T45, X10, T115, T22, T9) → QS53_IN_AA(T45, X10)
P52_IN_AAAGA(T45, T120, T115, T22, T9) → U17_AAAGA(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_AAAGA(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
U17_AAAGA(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → APP54_IN_AGAA(T115, T22, T120, T9)
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → U15_AGAA(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)
QS1_IN_AA(.(T265, []), T9) → U31_AA(T265, T9, qs99_in_a(X9))
QS1_IN_AA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_AA(.(T265, []), T9) → U32_AA(T265, T9, qs99_in_a(T268))
U32_AA(T265, T9, qs99_out_a(T268)) → U33_AA(T265, T9, qs99_in_a(X10))
U32_AA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_AA(T265, T9, qs99_out_a(T268)) → U34_AA(T265, T9, T268, qs99_in_a(T273))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → U35_AA(T265, T9, app54_in_gaga(T268, T265, T273, T9))
U34_AA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GAGA(T268, T265, T273, T9)
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GAGA(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_AA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_AA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_AA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
U21_GA(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_GA(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_GA(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U23_GA(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → P52_IN_AAAGA(T45, X10, T115, T22, T9)
QS1_IN_GA(.(T239, .(T240, T241)), T9) → U25_GA(T239, T240, T241, T9, le33_in_gg(T239, T240))
QS1_IN_GA(.(T239, .(T240, T241)), T9) → LE33_IN_GG(T239, T240)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → PART20_IN_GGAA(T239, T241, X437, X438)
U25_GA(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_GA(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_GA(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → QS53_IN_AA(T249, X9)
U27_GA(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_GA(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_GA(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U29_GA(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → P52_IN_AAAGA(.(T240, T250), X10, T256, T239, T9)
QS1_IN_GA(.(T265, []), T9) → U31_GA(T265, T9, qs99_in_a(X9))
QS1_IN_GA(.(T265, []), T9) → QS99_IN_A(X9)
QS1_IN_GA(.(T265, []), T9) → U32_GA(T265, T9, qs99_in_a(T268))
U32_GA(T265, T9, qs99_out_a(T268)) → U33_GA(T265, T9, qs99_in_a(X10))
U32_GA(T265, T9, qs99_out_a(T268)) → QS99_IN_A(X10)
U32_GA(T265, T9, qs99_out_a(T268)) → U34_GA(T265, T9, T268, qs99_in_a(T273))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → U35_GA(T265, T9, app54_in_ggga(T268, T265, T273, T9))
U34_GA(T265, T9, T268, qs99_out_a(T273)) → APP54_IN_GGGA(T268, T265, T273, T9)
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → U15_GGGA(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x3, x5)
GT10_IN_GG(x1, x2)  =  GT10_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x3, x6)
LE33_IN_GG(x1, x2)  =  LE33_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x6)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x5)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(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(x3)
U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x6)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)
U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x1, x5)
U28_AA(x1, x2, x3, x4, x5)  =  U28_AA(x5)
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U7_AA(x1, x2, x3, x4)  =  U7_AA(x4)
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)
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4)  =  U11_AA(x4)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4)  =  U13_AA(x4)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA
U14_AAAA(x1, x2, x3, x4, x5, x6)  =  U14_AAAA(x6)
U29_AA(x1, x2, x3, x4, x5, x6)  =  U29_AA(x1, x6)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
P52_IN_AAAGA(x1, x2, x3, x4, x5)  =  P52_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x6)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)
U15_AGAA(x1, x2, x3, x4, x5, x6)  =  U15_AGAA(x6)
U31_AA(x1, x2, x3)  =  U31_AA(x3)
QS99_IN_A(x1)  =  QS99_IN_A
U32_AA(x1, x2, x3)  =  U32_AA(x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x3, x4)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)
U15_GAGA(x1, x2, x3, x4, x5, x6)  =  U15_GAGA(x6)
U23_AA(x1, x2, x3, x4, x5, x6)  =  U23_AA(x1, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U23_GA(x1, x2, x3, x4, x5, x6)  =  U23_GA(x1, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x5)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x3, x5)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_GA(x5)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x5)
U31_GA(x1, x2, x3)  =  U31_GA(x3)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x3)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x1, x3, x4)
U35_GA(x1, x2, x3)  =  U35_GA(x3)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)
U15_GGGA(x1, x2, x3, x4, x5, x6)  =  U15_GGGA(x1, x6)

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

(137) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 91 less nodes.

(138) Complex Obligation (AND)

(139) Obligation:

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

APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)

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

(140) UsableRulesProof (EQUIVALENT transformation)

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

(141) Obligation:

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

APP54_IN_GGGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GGGA(T211, T212, T213, T215)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP54_IN_GGGA(x1, x2, x3, x4)  =  APP54_IN_GGGA(x1, x2, x3)

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

(142) PiDPToQDPProof (SOUND transformation)

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

(143) Obligation:

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

APP54_IN_GGGA(.(T210, T211), T212, T213) → APP54_IN_GGGA(T211, T212, T213)

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

(144) 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:

• APP54_IN_GGGA(.(T210, T211), T212, T213) → APP54_IN_GGGA(T211, T212, T213)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(145) YES

(146) Obligation:

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

APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)

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

(147) UsableRulesProof (EQUIVALENT transformation)

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

(148) Obligation:

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

APP54_IN_GAGA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_GAGA(T211, T212, T213, T215)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP54_IN_GAGA(x1, x2, x3, x4)  =  APP54_IN_GAGA(x1, x3)

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

(149) PiDPToQDPProof (SOUND transformation)

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

(150) Obligation:

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

APP54_IN_GAGA(.(T210, T211), T213) → APP54_IN_GAGA(T211, T213)

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

(151) 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:

• APP54_IN_GAGA(.(T210, T211), T213) → APP54_IN_GAGA(T211, T213)
  The graph contains the following edges 1 > 1, 2 >= 2

(152) YES

(153) Obligation:

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

APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)

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

(154) UsableRulesProof (EQUIVALENT transformation)

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

(155) Obligation:

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

APP54_IN_AGAA(.(T210, T211), T212, T213, .(T210, T215)) → APP54_IN_AGAA(T211, T212, T213, T215)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP54_IN_AGAA(x1, x2, x3, x4)  =  APP54_IN_AGAA(x2)

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

(156) PiDPToQDPProof (SOUND transformation)

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

(157) Obligation:

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

APP54_IN_AGAA(T212) → APP54_IN_AGAA(T212)

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

(158) 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 = APP54_IN_AGAA(T212) evaluates to t =APP54_IN_AGAA(T212)

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 APP54_IN_AGAA(T212) to APP54_IN_AGAA(T212).

(159) NO

(160) Obligation:

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

APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA

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

(161) UsableRulesProof (EQUIVALENT transformation)

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

(162) Obligation:

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

APP65_IN_AAAA(.(T171, T172), T173, T174, .(T171, X354)) → APP65_IN_AAAA(T172, T173, T174, X354)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP65_IN_AAAA(x1, x2, x3, x4)  =  APP65_IN_AAAA

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

(163) PiDPToQDPProof (SOUND transformation)

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

(164) Obligation:

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

APP65_IN_AAAA → APP65_IN_AAAA

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

(165) 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 = APP65_IN_AAAA evaluates to t =APP65_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 APP65_IN_AAAA to APP65_IN_AAAA.

(166) NO

(167) Obligation:

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

LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
LE33_IN_AA(x1, x2)  =  LE33_IN_AA

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

(168) UsableRulesProof (EQUIVALENT transformation)

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

(169) Obligation:

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

LE33_IN_AA(s(T98), s(T99)) → LE33_IN_AA(T98, T99)

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

(170) PiDPToQDPProof (SOUND transformation)

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

(171) 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.

(172) 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.

(173) NO

(174) Obligation:

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

LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
LE33_IN_GA(x1, x2)  =  LE33_IN_GA(x1)

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

(175) UsableRulesProof (EQUIVALENT transformation)

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

(176) Obligation:

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

LE33_IN_GA(s(T98), s(T99)) → LE33_IN_GA(T98, T99)

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

(177) PiDPToQDPProof (SOUND transformation)

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

(178) Obligation:

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

LE33_IN_GA(s(T98)) → LE33_IN_GA(T98)

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

(179) 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(T98)) → LE33_IN_GA(T98)
  The graph contains the following edges 1 > 1

(180) YES

(181) Obligation:

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

GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
GT10_IN_GA(x1, x2)  =  GT10_IN_GA(x1)

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

(182) UsableRulesProof (EQUIVALENT transformation)

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

(183) Obligation:

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

GT10_IN_GA(s(T37), s(T38)) → GT10_IN_GA(T37, T38)

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

(184) PiDPToQDPProof (SOUND transformation)

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

(185) Obligation:

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

GT10_IN_GA(s(T37)) → GT10_IN_GA(T37)

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

(186) 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(T37)) → GT10_IN_GA(T37)
  The graph contains the following edges 1 > 1

(187) YES

(188) Obligation:

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

U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)

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

(189) UsableRulesProof (EQUIVALENT transformation)

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

(190) Obligation:

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

U2_GAAA(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → PART20_IN_GAAA(T61, T63, X119, X120)
PART20_IN_GAAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GAAA(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
PART20_IN_GAAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GAAA(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
U4_GAAA(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → PART20_IN_GAAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
gt10_in_ga(x1, x2)  =  gt10_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
gt10_out_ga(x1, x2)  =  gt10_out_ga(x2)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
PART20_IN_GAAA(x1, x2, x3, x4)  =  PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)

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

(191) PiDPToQDPProof (SOUND transformation)

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

(192) Obligation:

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

U2_GAAA(T61, gt10_out_ga(T62)) → PART20_IN_GAAA(T61)
PART20_IN_GAAA(T61) → U2_GAAA(T61, gt10_in_ga(T61))
PART20_IN_GAAA(T83) → U4_GAAA(T83, le33_in_ga(T83))
U4_GAAA(T83, le33_out_ga) → PART20_IN_GAAA(T83)

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(193) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T61) → U2_GAAA(T61, gt10_in_ga(T61)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0))

(194) Obligation:

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

U2_GAAA(T61, gt10_out_ga(T62)) → PART20_IN_GAAA(T61)
PART20_IN_GAAA(T83) → U4_GAAA(T83, le33_in_ga(T83))
U4_GAAA(T83, le33_out_ga) → PART20_IN_GAAA(T83)
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(195) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T83) → U4_GAAA(T83, le33_in_ga(T83)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga)

(196) Obligation:

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

U2_GAAA(T61, gt10_out_ga(T62)) → PART20_IN_GAAA(T61)
U4_GAAA(T83, le33_out_ga) → PART20_IN_GAAA(T83)
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga)

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(197) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GAAA(T61, gt10_out_ga(T62)) → PART20_IN_GAAA(T61) we obtained the following new rules [LPAR04]:

U2_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
U2_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))

(198) Obligation:

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

U4_GAAA(T83, le33_out_ga) → PART20_IN_GAAA(T83)
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga)
U2_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
U2_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(199) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GAAA(T83, le33_out_ga) → PART20_IN_GAAA(T83) we obtained the following new rules [LPAR04]:

U4_GAAA(s(z0), le33_out_ga) → PART20_IN_GAAA(s(z0))
U4_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)

(200) Obligation:

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

PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(le33_in_ga(x0)))
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga)
U2_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
U2_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))
U4_GAAA(s(z0), le33_out_ga) → PART20_IN_GAAA(s(z0))
U4_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(201) DependencyGraphProof (EQUIVALENT transformation)

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

(202) Complex Obligation (AND)

(203) Obligation:

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

U4_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga)

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(204) 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.

(205) Obligation:

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

U4_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U4_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)
U6_ga(x0)

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

(206) 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)
U6_ga(x0)

(207) Obligation:

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

U4_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga)

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

(208) 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) → U4_GAAA(0, le33_out_ga)
with rule PART20_IN_GAAA(0) → U4_GAAA(0, le33_out_ga) at position [] and matcher [ ]

U4_GAAA(0, le33_out_ga) → PART20_IN_GAAA(0)
with rule U4_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.

(209) NO

(210) Obligation:

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

U2_GAAA(s(z0), gt10_out_ga(x1)) → PART20_IN_GAAA(s(z0))
PART20_IN_GAAA(s(x0)) → U2_GAAA(s(x0), U1_ga(gt10_in_ga(x0)))
U2_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))
PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0))
PART20_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U6_ga(le33_in_ga(x0)))
U4_GAAA(s(z0), le33_out_ga) → PART20_IN_GAAA(s(z0))

The TRS R consists of the following rules:

gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_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)
U6_ga(x0)

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

(211) 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)) → U2_GAAA(s(0), gt10_out_ga(0))
with rule PART20_IN_GAAA(s(0)) → U2_GAAA(s(0), gt10_out_ga(0)) at position [] and matcher [ ]

U2_GAAA(s(0), gt10_out_ga(0)) → PART20_IN_GAAA(s(0))
with rule U2_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.

(212) NO

(213) Obligation:

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

GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
GT10_IN_AA(x1, x2)  =  GT10_IN_AA

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

(214) UsableRulesProof (EQUIVALENT transformation)

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

(215) Obligation:

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

GT10_IN_AA(s(T37), s(T38)) → GT10_IN_AA(T37, T38)

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

(216) PiDPToQDPProof (SOUND transformation)

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

(217) 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.

(218) 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.

(219) NO

(220) Obligation:

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

QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)

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

(221) UsableRulesProof (EQUIVALENT transformation)

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

(222) Obligation:

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

QS53_IN_AA(.(T131, T132), X275) → U8_AA(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → QS53_IN_AA(T136, X273)
U8_AA(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_AA(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_AA(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → QS53_IN_AA(T137, X274)

The TRS R consists of the following rules:

part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
qs53_in_aa([], []) → qs53_out_aa([], [])
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
[]  =  []
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)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
QS53_IN_AA(x1, x2)  =  QS53_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)

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

(223) PiDPToQDPProof (SOUND transformation)

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

(224) Obligation:

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

QS53_IN_AA → U8_AA(part20_in_aaaa)
U8_AA(part20_out_aaaa) → QS53_IN_AA
U8_AA(part20_out_aaaa) → U10_AA(qs53_in_aa)
U10_AA(qs53_out_aa) → QS53_IN_AA

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U4_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs53_in_aa → U7_aa(part20_in_aaaa)
qs53_in_aa → U8_aa(part20_in_aaaa)
qs53_in_aa → qs53_out_aa
U2_aaaa(gt10_out_aa(T61, T62)) → part20_out_aaaa
U2_aaaa(gt10_out_aa(T61, T62)) → U3_aaaa(part20_in_gaaa(T61))
U4_aaaa(le33_out_aa(T83)) → part20_out_aaaa
U4_aaaa(le33_out_aa(T83)) → U5_aaaa(part20_in_gaaa(T83))
U7_aa(part20_out_aaaa) → qs53_out_aa
U8_aa(part20_out_aaaa) → U9_aa(qs53_in_aa)
U8_aa(part20_out_aaaa) → U10_aa(qs53_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U3_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U6_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U5_aaaa(part20_out_gaaa) → part20_out_aaaa
U9_aa(qs53_out_aa) → qs53_out_aa
U10_aa(qs53_out_aa) → U11_aa(qs53_in_aa)
U10_aa(qs53_out_aa) → U12_aa(qs53_in_aa)
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa
U6_aa(le33_out_aa(T98)) → le33_out_aa(s(T98))
U11_aa(qs53_out_aa) → qs53_out_aa
U12_aa(qs53_out_aa) → U13_aa(app65_in_aaaa)
U2_gaaa(T61, gt10_out_ga(T62)) → part20_out_gaaa
U2_gaaa(T61, gt10_out_ga(T62)) → U3_gaaa(part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga) → part20_out_gaaa
U4_gaaa(T83, le33_out_ga) → U5_gaaa(part20_in_gaaa(T83))
U13_aa(app65_out_aaaa) → qs53_out_aa
gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U3_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U5_gaaa(part20_out_gaaa) → part20_out_gaaa
app65_in_aaaa → U14_aaaa(app65_in_aaaa)
app65_in_aaaa → app65_out_aaaa
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_ga(le33_out_ga) → le33_out_ga
U14_aaaa(app65_out_aaaa) → app65_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs53_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
gt10_in_aa
U3_aaaa(x0)
le33_in_aa
U5_aaaa(x0)
U9_aa(x0)
U10_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U6_aa(x0)
U11_aa(x0)
U12_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U13_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0)
le33_in_ga(x0)
U5_gaaa(x0)
app65_in_aaaa
U1_ga(x0)
U6_ga(x0)
U14_aaaa(x0)

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

(225) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS53_IN_AA → U8_AA(part20_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS53_IN_AA → U8_AA(U2_aaaa(gt10_in_aa))
QS53_IN_AA → U8_AA(U4_aaaa(le33_in_aa))
QS53_IN_AA → U8_AA(part20_out_aaaa)

(226) Obligation:

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

U8_AA(part20_out_aaaa) → QS53_IN_AA
U8_AA(part20_out_aaaa) → U10_AA(qs53_in_aa)
U10_AA(qs53_out_aa) → QS53_IN_AA
QS53_IN_AA → U8_AA(U2_aaaa(gt10_in_aa))
QS53_IN_AA → U8_AA(U4_aaaa(le33_in_aa))
QS53_IN_AA → U8_AA(part20_out_aaaa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U4_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs53_in_aa → U7_aa(part20_in_aaaa)
qs53_in_aa → U8_aa(part20_in_aaaa)
qs53_in_aa → qs53_out_aa
U2_aaaa(gt10_out_aa(T61, T62)) → part20_out_aaaa
U2_aaaa(gt10_out_aa(T61, T62)) → U3_aaaa(part20_in_gaaa(T61))
U4_aaaa(le33_out_aa(T83)) → part20_out_aaaa
U4_aaaa(le33_out_aa(T83)) → U5_aaaa(part20_in_gaaa(T83))
U7_aa(part20_out_aaaa) → qs53_out_aa
U8_aa(part20_out_aaaa) → U9_aa(qs53_in_aa)
U8_aa(part20_out_aaaa) → U10_aa(qs53_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U3_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U6_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U5_aaaa(part20_out_gaaa) → part20_out_aaaa
U9_aa(qs53_out_aa) → qs53_out_aa
U10_aa(qs53_out_aa) → U11_aa(qs53_in_aa)
U10_aa(qs53_out_aa) → U12_aa(qs53_in_aa)
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa
U6_aa(le33_out_aa(T98)) → le33_out_aa(s(T98))
U11_aa(qs53_out_aa) → qs53_out_aa
U12_aa(qs53_out_aa) → U13_aa(app65_in_aaaa)
U2_gaaa(T61, gt10_out_ga(T62)) → part20_out_gaaa
U2_gaaa(T61, gt10_out_ga(T62)) → U3_gaaa(part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga) → part20_out_gaaa
U4_gaaa(T83, le33_out_ga) → U5_gaaa(part20_in_gaaa(T83))
U13_aa(app65_out_aaaa) → qs53_out_aa
gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U3_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U5_gaaa(part20_out_gaaa) → part20_out_gaaa
app65_in_aaaa → U14_aaaa(app65_in_aaaa)
app65_in_aaaa → app65_out_aaaa
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_ga(le33_out_ga) → le33_out_ga
U14_aaaa(app65_out_aaaa) → app65_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs53_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
gt10_in_aa
U3_aaaa(x0)
le33_in_aa
U5_aaaa(x0)
U9_aa(x0)
U10_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U6_aa(x0)
U11_aa(x0)
U12_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U13_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0)
le33_in_ga(x0)
U5_gaaa(x0)
app65_in_aaaa
U1_ga(x0)
U6_ga(x0)
U14_aaaa(x0)

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

(227) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U8_AA(part20_out_aaaa) → U10_AA(qs53_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U8_AA(part20_out_aaaa) → U10_AA(U7_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(U8_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(qs53_out_aa)

(228) Obligation:

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

U8_AA(part20_out_aaaa) → QS53_IN_AA
U10_AA(qs53_out_aa) → QS53_IN_AA
QS53_IN_AA → U8_AA(U2_aaaa(gt10_in_aa))
QS53_IN_AA → U8_AA(U4_aaaa(le33_in_aa))
QS53_IN_AA → U8_AA(part20_out_aaaa)
U8_AA(part20_out_aaaa) → U10_AA(U7_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(U8_aa(part20_in_aaaa))
U8_AA(part20_out_aaaa) → U10_AA(qs53_out_aa)

The TRS R consists of the following rules:

part20_in_aaaa → U2_aaaa(gt10_in_aa)
part20_in_aaaa → U4_aaaa(le33_in_aa)
part20_in_aaaa → part20_out_aaaa
qs53_in_aa → U7_aa(part20_in_aaaa)
qs53_in_aa → U8_aa(part20_in_aaaa)
qs53_in_aa → qs53_out_aa
U2_aaaa(gt10_out_aa(T61, T62)) → part20_out_aaaa
U2_aaaa(gt10_out_aa(T61, T62)) → U3_aaaa(part20_in_gaaa(T61))
U4_aaaa(le33_out_aa(T83)) → part20_out_aaaa
U4_aaaa(le33_out_aa(T83)) → U5_aaaa(part20_in_gaaa(T83))
U7_aa(part20_out_aaaa) → qs53_out_aa
U8_aa(part20_out_aaaa) → U9_aa(qs53_in_aa)
U8_aa(part20_out_aaaa) → U10_aa(qs53_in_aa)
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U3_aaaa(part20_out_gaaa) → part20_out_aaaa
le33_in_aa → U6_aa(le33_in_aa)
le33_in_aa → le33_out_aa(0)
U5_aaaa(part20_out_gaaa) → part20_out_aaaa
U9_aa(qs53_out_aa) → qs53_out_aa
U10_aa(qs53_out_aa) → U11_aa(qs53_in_aa)
U10_aa(qs53_out_aa) → U12_aa(qs53_in_aa)
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa
U6_aa(le33_out_aa(T98)) → le33_out_aa(s(T98))
U11_aa(qs53_out_aa) → qs53_out_aa
U12_aa(qs53_out_aa) → U13_aa(app65_in_aaaa)
U2_gaaa(T61, gt10_out_ga(T62)) → part20_out_gaaa
U2_gaaa(T61, gt10_out_ga(T62)) → U3_gaaa(part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga) → part20_out_gaaa
U4_gaaa(T83, le33_out_ga) → U5_gaaa(part20_in_gaaa(T83))
U13_aa(app65_out_aaaa) → qs53_out_aa
gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U3_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U5_gaaa(part20_out_gaaa) → part20_out_gaaa
app65_in_aaaa → U14_aaaa(app65_in_aaaa)
app65_in_aaaa → app65_out_aaaa
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_ga(le33_out_ga) → le33_out_ga
U14_aaaa(app65_out_aaaa) → app65_out_aaaa

The set Q consists of the following terms:

part20_in_aaaa
qs53_in_aa
U2_aaaa(x0)
U4_aaaa(x0)
U7_aa(x0)
U8_aa(x0)
gt10_in_aa
U3_aaaa(x0)
le33_in_aa
U5_aaaa(x0)
U9_aa(x0)
U10_aa(x0)
U1_aa(x0)
part20_in_gaaa(x0)
U6_aa(x0)
U11_aa(x0)
U12_aa(x0)
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U13_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0)
le33_in_ga(x0)
U5_gaaa(x0)
app65_in_aaaa
U1_ga(x0)
U6_ga(x0)
U14_aaaa(x0)

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

(229) 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 = QS53_IN_AA evaluates to t =QS53_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS53_IN_AA → U8_AA(part20_out_aaaa)
with rule QS53_IN_AA → U8_AA(part20_out_aaaa) at position [] and matcher [ ]

U8_AA(part20_out_aaaa) → QS53_IN_AA
with rule U8_AA(part20_out_aaaa) → QS53_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.

(230) NO

(231) Obligation:

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

U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)

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

(232) UsableRulesProof (EQUIVALENT transformation)

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

(233) Obligation:

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

U19_AA(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_AA(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_AA(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → QS1_IN_AA(.(T23, T44), X9)
QS1_IN_AA(.(T22, .(T23, T24)), T9) → U19_AA(T22, T23, T24, T9, gt10_in_aa(T22, T23))

The TRS R consists of the following rules:

part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
[]  =  []
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)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)
U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x1, x5)

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

(234) PiDPToQDPProof (SOUND transformation)

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

(235) Obligation:

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

U19_AA(gt10_out_aa(T22, T23)) → U21_AA(T22, part20_in_gaaa(T22))
U21_AA(T22, part20_out_gaaa) → QS1_IN_AA
QS1_IN_AA → U19_AA(gt10_in_aa)

The TRS R consists of the following rules:

part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U2_gaaa(T61, gt10_out_ga(T62)) → part20_out_gaaa
U2_gaaa(T61, gt10_out_ga(T62)) → U3_gaaa(part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga) → part20_out_gaaa
U4_gaaa(T83, le33_out_ga) → U5_gaaa(part20_in_gaaa(T83))
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U3_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U5_gaaa(part20_out_gaaa) → part20_out_gaaa
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

part20_in_gaaa(x0)
gt10_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0)
le33_in_ga(x0)
U5_gaaa(x0)
U1_ga(x0)
U6_ga(x0)

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

(236) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U19_AA(gt10_out_aa(T22, T23)) → U21_AA(T22, part20_in_gaaa(T22)) at position [1] we obtained the following new rules [LPAR04]:

U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U2_gaaa(x0, gt10_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U4_gaaa(x0, le33_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa)

(237) Obligation:

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

U21_AA(T22, part20_out_gaaa) → QS1_IN_AA
QS1_IN_AA → U19_AA(gt10_in_aa)
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U2_gaaa(x0, gt10_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U4_gaaa(x0, le33_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa)

The TRS R consists of the following rules:

part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U2_gaaa(T61, gt10_out_ga(T62)) → part20_out_gaaa
U2_gaaa(T61, gt10_out_ga(T62)) → U3_gaaa(part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga) → part20_out_gaaa
U4_gaaa(T83, le33_out_ga) → U5_gaaa(part20_in_gaaa(T83))
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U3_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U5_gaaa(part20_out_gaaa) → part20_out_gaaa
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

part20_in_gaaa(x0)
gt10_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0)
le33_in_ga(x0)
U5_gaaa(x0)
U1_ga(x0)
U6_ga(x0)

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

(238) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS1_IN_AA → U19_AA(gt10_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QS1_IN_AA → U19_AA(U1_aa(gt10_in_aa))
QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0))

(239) Obligation:

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

U21_AA(T22, part20_out_gaaa) → QS1_IN_AA
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U2_gaaa(x0, gt10_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, U4_gaaa(x0, le33_in_ga(x0)))
U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa)
QS1_IN_AA → U19_AA(U1_aa(gt10_in_aa))
QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0))

The TRS R consists of the following rules:

part20_in_gaaa(T61) → U2_gaaa(T61, gt10_in_ga(T61))
part20_in_gaaa(T83) → U4_gaaa(T83, le33_in_ga(T83))
part20_in_gaaa(T112) → part20_out_gaaa
gt10_in_aa → U1_aa(gt10_in_aa)
gt10_in_aa → gt10_out_aa(s(0), 0)
U2_gaaa(T61, gt10_out_ga(T62)) → part20_out_gaaa
U2_gaaa(T61, gt10_out_ga(T62)) → U3_gaaa(part20_in_gaaa(T61))
U4_gaaa(T83, le33_out_ga) → part20_out_gaaa
U4_gaaa(T83, le33_out_ga) → U5_gaaa(part20_in_gaaa(T83))
U1_aa(gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
gt10_in_ga(s(T37)) → U1_ga(gt10_in_ga(T37))
gt10_in_ga(s(0)) → gt10_out_ga(0)
U3_gaaa(part20_out_gaaa) → part20_out_gaaa
le33_in_ga(s(T98)) → U6_ga(le33_in_ga(T98))
le33_in_ga(0) → le33_out_ga
U5_gaaa(part20_out_gaaa) → part20_out_gaaa
U1_ga(gt10_out_ga(T38)) → gt10_out_ga(s(T38))
U6_ga(le33_out_ga) → le33_out_ga

The set Q consists of the following terms:

part20_in_gaaa(x0)
gt10_in_aa
U2_gaaa(x0, x1)
U4_gaaa(x0, x1)
U1_aa(x0)
gt10_in_ga(x0)
U3_gaaa(x0)
le33_in_ga(x0)
U5_gaaa(x0)
U1_ga(x0)
U6_ga(x0)

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

(240) NonTerminationProof (EQUIVALENT transformation)

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

s = QS1_IN_AA evaluates to t =QS1_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0))
with rule QS1_IN_AA → U19_AA(gt10_out_aa(s(0), 0)) at position [] and matcher [ ]

U19_AA(gt10_out_aa(s(0), 0)) → U21_AA(s(0), part20_out_gaaa)
with rule U19_AA(gt10_out_aa(x0, y1)) → U21_AA(x0, part20_out_gaaa) at position [] and matcher [x0 / s(0), y1 / 0]

U21_AA(s(0), part20_out_gaaa) → QS1_IN_AA
with rule U21_AA(T22, part20_out_gaaa) → QS1_IN_AA

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


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

(241) NO

(242) Obligation:

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

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
LE33_IN_GG(x1, x2)  =  LE33_IN_GG(x1, x2)

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

(243) UsableRulesProof (EQUIVALENT transformation)

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

(244) Obligation:

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

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

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

(245) PiDPToQDPProof (EQUIVALENT transformation)

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

(246) Obligation:

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

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

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

(247) 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_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
  The graph contains the following edges 1 > 1, 2 > 2

(248) YES

(249) Obligation:

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

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
GT10_IN_GG(x1, x2)  =  GT10_IN_GG(x1, x2)

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

(250) UsableRulesProof (EQUIVALENT transformation)

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

(251) Obligation:

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

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

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

(252) PiDPToQDPProof (EQUIVALENT transformation)

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

(253) Obligation:

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

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

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

(254) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
  The graph contains the following edges 1 > 1, 2 > 2

(255) YES

(256) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

qs1_in_ga(.(T22, .(T23, T24)), T9) → U19_ga(T22, T23, T24, T9, gt10_in_gg(T22, T23))
gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U20_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
part20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U2_ggaa(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U2_ggaa(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → U3_ggaa(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
part20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_ggaa(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U4_ggaa(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → U5_ggaa(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
part20_in_ggaa(T112, [], [], []) → part20_out_ggaa(T112, [], [], [])
U5_ggaa(T83, T84, T85, X170, X171, part20_out_ggaa(T83, T85, X170, X171)) → part20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U3_ggaa(T61, T62, T63, X119, X120, part20_out_ggaa(T61, T63, X119, X120)) → part20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U20_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, X47, X48)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U19_ga(T22, T23, T24, T9, gt10_out_gg(T22, T23)) → U21_ga(T22, T23, T24, T9, part20_in_ggaa(T22, T24, T44, T45))
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U22_ga(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T22, .(T23, T24)), T9) → U19_aa(T22, T23, T24, T9, gt10_in_aa(T22, T23))
gt10_in_aa(s(T37), s(T38)) → U1_aa(T37, T38, gt10_in_aa(T37, T38))
gt10_in_aa(s(0), 0) → gt10_out_aa(s(0), 0)
U1_aa(T37, T38, gt10_out_aa(T37, T38)) → gt10_out_aa(s(T37), s(T38))
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U20_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, X47, X48))
part20_in_gaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_gaaa(T61, T62, T63, X119, X120, gt10_in_ga(T61, T62))
gt10_in_ga(s(T37), s(T38)) → U1_ga(T37, T38, gt10_in_ga(T37, T38))
gt10_in_ga(s(0), 0) → gt10_out_ga(s(0), 0)
U1_ga(T37, T38, gt10_out_ga(T37, T38)) → gt10_out_ga(s(T37), s(T38))
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_gaaa(T61, T62, T63, X119, X120, gt10_out_ga(T61, T62)) → U3_gaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
part20_in_gaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_gaaa(T83, T84, T85, X170, X171, le33_in_ga(T83, T84))
le33_in_ga(s(T98), s(T99)) → U6_ga(T98, T99, le33_in_ga(T98, T99))
le33_in_ga(0, s(T106)) → le33_out_ga(0, s(T106))
le33_in_ga(0, 0) → le33_out_ga(0, 0)
U6_ga(T98, T99, le33_out_ga(T98, T99)) → le33_out_ga(s(T98), s(T99))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_gaaa(T83, T84, T85, X170, X171, le33_out_ga(T83, T84)) → U5_gaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
part20_in_gaaa(T112, [], [], []) → part20_out_gaaa(T112, [], [], [])
U5_gaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_gaaa(T83, .(T84, T85), X170, .(T84, X171))
U3_gaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_gaaa(T61, .(T62, T63), .(T62, X119), X120)
U20_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, X47, X48)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U19_aa(T22, T23, T24, T9, gt10_out_aa(T22, T23)) → U21_aa(T22, T23, T24, T9, part20_in_gaaa(T22, T24, T44, T45))
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U22_aa(T22, T23, T24, T9, qs1_in_aa(.(T23, T44), X9))
qs1_in_aa(.(T239, .(T240, T241)), T9) → U25_aa(T239, T240, T241, T9, le33_in_aa(T239, T240))
le33_in_aa(s(T98), s(T99)) → U6_aa(T98, T99, le33_in_aa(T98, T99))
le33_in_aa(0, s(T106)) → le33_out_aa(0, s(T106))
le33_in_aa(0, 0) → le33_out_aa(0, 0)
U6_aa(T98, T99, le33_out_aa(T98, T99)) → le33_out_aa(s(T98), s(T99))
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U26_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, X437, X438))
U26_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, X437, X438)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U25_aa(T239, T240, T241, T9, le33_out_aa(T239, T240)) → U27_aa(T239, T240, T241, T9, part20_in_gaaa(T239, T241, T249, T250))
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U28_aa(T239, T240, T241, T9, qs53_in_aa(T249, X9))
qs53_in_aa(.(T131, T132), X275) → U7_aa(T131, T132, X275, part20_in_aaaa(T131, T132, X271, X272))
part20_in_aaaa(T61, .(T62, T63), .(T62, X119), X120) → U2_aaaa(T61, T62, T63, X119, X120, gt10_in_aa(T61, T62))
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
U2_aaaa(T61, T62, T63, X119, X120, gt10_out_aa(T61, T62)) → U3_aaaa(T61, T62, T63, X119, X120, part20_in_gaaa(T61, T63, X119, X120))
U3_aaaa(T61, T62, T63, X119, X120, part20_out_gaaa(T61, T63, X119, X120)) → part20_out_aaaa(T61, .(T62, T63), .(T62, X119), X120)
part20_in_aaaa(T83, .(T84, T85), X170, .(T84, X171)) → U4_aaaa(T83, T84, T85, X170, X171, le33_in_aa(T83, T84))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
U4_aaaa(T83, T84, T85, X170, X171, le33_out_aa(T83, T84)) → U5_aaaa(T83, T84, T85, X170, X171, part20_in_gaaa(T83, T85, X170, X171))
U5_aaaa(T83, T84, T85, X170, X171, part20_out_gaaa(T83, T85, X170, X171)) → part20_out_aaaa(T83, .(T84, T85), X170, .(T84, X171))
part20_in_aaaa(T112, [], [], []) → part20_out_aaaa(T112, [], [], [])
U7_aa(T131, T132, X275, part20_out_aaaa(T131, T132, X271, X272)) → qs53_out_aa(.(T131, T132), X275)
qs53_in_aa(.(T131, T132), X275) → U8_aa(T131, T132, X275, part20_in_aaaa(T131, T132, T136, T137))
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U9_aa(T131, T132, X275, qs53_in_aa(T136, X273))
qs53_in_aa([], []) → qs53_out_aa([], [])
U9_aa(T131, T132, X275, qs53_out_aa(T136, X273)) → qs53_out_aa(.(T131, T132), X275)
U8_aa(T131, T132, X275, part20_out_aaaa(T131, T132, T136, T137)) → U10_aa(T131, T132, X275, T137, qs53_in_aa(T136, T143))
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U11_aa(T131, T132, X275, qs53_in_aa(T137, X274))
U11_aa(T131, T132, X275, qs53_out_aa(T137, X274)) → qs53_out_aa(.(T131, T132), X275)
U10_aa(T131, T132, X275, T137, qs53_out_aa(T136, T143)) → U12_aa(T131, T132, X275, T143, qs53_in_aa(T137, T148))
U12_aa(T131, T132, X275, T143, qs53_out_aa(T137, T148)) → U13_aa(T131, T132, X275, app65_in_aaaa(T143, T131, T148, X275))
app65_in_aaaa(.(T171, T172), T173, T174, .(T171, X354)) → U14_aaaa(T171, T172, T173, T174, X354, app65_in_aaaa(T172, T173, T174, X354))
app65_in_aaaa([], T183, T184, .(T183, T184)) → app65_out_aaaa([], T183, T184, .(T183, T184))
U14_aaaa(T171, T172, T173, T174, X354, app65_out_aaaa(T172, T173, T174, X354)) → app65_out_aaaa(.(T171, T172), T173, T174, .(T171, X354))
U13_aa(T131, T132, X275, app65_out_aaaa(T143, T131, T148, X275)) → qs53_out_aa(.(T131, T132), X275)
U28_aa(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
U27_aa(T239, T240, T241, T9, part20_out_gaaa(T239, T241, T249, T250)) → U29_aa(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_aa(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_aa(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
p52_in_aaaga(T45, X10, T115, T22, T9) → U16_aaaga(T45, X10, T115, T22, T9, qs53_in_aa(T45, X10))
U16_aaaga(T45, X10, T115, T22, T9, qs53_out_aa(T45, X10)) → p52_out_aaaga(T45, X10, T115, T22, T9)
p52_in_aaaga(T45, T120, T115, T22, T9) → U17_aaaga(T45, T120, T115, T22, T9, qs53_in_aa(T45, T120))
U17_aaaga(T45, T120, T115, T22, T9, qs53_out_aa(T45, T120)) → U18_aaaga(T45, T120, T115, T22, T9, app54_in_agaa(T115, T22, T120, T9))
app54_in_agaa(.(T210, T211), T212, T213, .(T210, T215)) → U15_agaa(T210, T211, T212, T213, T215, app54_in_agaa(T211, T212, T213, T215))
app54_in_agaa([], T225, T226, .(T225, T226)) → app54_out_agaa([], T225, T226, .(T225, T226))
U15_agaa(T210, T211, T212, T213, T215, app54_out_agaa(T211, T212, T213, T215)) → app54_out_agaa(.(T210, T211), T212, T213, .(T210, T215))
U18_aaaga(T45, T120, T115, T22, T9, app54_out_agaa(T115, T22, T120, T9)) → p52_out_aaaga(T45, T120, T115, T22, T9)
U30_aa(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_aa(.(T239, .(T240, T241)), T9)
qs1_in_aa(.(T265, []), T9) → U31_aa(T265, T9, qs99_in_a(X9))
qs99_in_a([]) → qs99_out_a([])
U31_aa(T265, T9, qs99_out_a(X9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa(.(T265, []), T9) → U32_aa(T265, T9, qs99_in_a(T268))
U32_aa(T265, T9, qs99_out_a(T268)) → U33_aa(T265, T9, qs99_in_a(X10))
U33_aa(T265, T9, qs99_out_a(X10)) → qs1_out_aa(.(T265, []), T9)
U32_aa(T265, T9, qs99_out_a(T268)) → U34_aa(T265, T9, T268, qs99_in_a(T273))
U34_aa(T265, T9, T268, qs99_out_a(T273)) → U35_aa(T265, T9, app54_in_gaga(T268, T265, T273, T9))
app54_in_gaga(.(T210, T211), T212, T213, .(T210, T215)) → U15_gaga(T210, T211, T212, T213, T215, app54_in_gaga(T211, T212, T213, T215))
app54_in_gaga([], T225, T226, .(T225, T226)) → app54_out_gaga([], T225, T226, .(T225, T226))
U15_gaga(T210, T211, T212, T213, T215, app54_out_gaga(T211, T212, T213, T215)) → app54_out_gaga(.(T210, T211), T212, T213, .(T210, T215))
U35_aa(T265, T9, app54_out_gaga(T268, T265, T273, T9)) → qs1_out_aa(.(T265, []), T9)
qs1_in_aa([], []) → qs1_out_aa([], [])
U22_aa(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U21_aa(T22, T23, T24, T9, part20_out_gaaa(T22, T24, T44, T45)) → U23_aa(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_aa(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_aa(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_aa(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_aa(.(T22, .(T23, T24)), T9)
U22_ga(T22, T23, T24, T9, qs1_out_aa(.(T23, T44), X9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
U21_ga(T22, T23, T24, T9, part20_out_ggaa(T22, T24, T44, T45)) → U23_ga(T22, T23, T24, T9, T45, qs1_in_aa(.(T23, T44), T115))
U23_ga(T22, T23, T24, T9, T45, qs1_out_aa(.(T23, T44), T115)) → U24_ga(T22, T23, T24, T9, p52_in_aaaga(T45, X10, T115, T22, T9))
U24_ga(T22, T23, T24, T9, p52_out_aaaga(T45, X10, T115, T22, T9)) → qs1_out_ga(.(T22, .(T23, T24)), T9)
qs1_in_ga(.(T239, .(T240, T241)), T9) → U25_ga(T239, T240, T241, T9, le33_in_gg(T239, T240))
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U26_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, X437, X438))
U26_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, X437, X438)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U25_ga(T239, T240, T241, T9, le33_out_gg(T239, T240)) → U27_ga(T239, T240, T241, T9, part20_in_ggaa(T239, T241, T249, T250))
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U28_ga(T239, T240, T241, T9, qs53_in_aa(T249, X9))
U28_ga(T239, T240, T241, T9, qs53_out_aa(T249, X9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
U27_ga(T239, T240, T241, T9, part20_out_ggaa(T239, T241, T249, T250)) → U29_ga(T239, T240, T241, T9, T250, qs53_in_aa(T249, T256))
U29_ga(T239, T240, T241, T9, T250, qs53_out_aa(T249, T256)) → U30_ga(T239, T240, T241, T9, p52_in_aaaga(.(T240, T250), X10, T256, T239, T9))
U30_ga(T239, T240, T241, T9, p52_out_aaaga(.(T240, T250), X10, T256, T239, T9)) → qs1_out_ga(.(T239, .(T240, T241)), T9)
qs1_in_ga(.(T265, []), T9) → U31_ga(T265, T9, qs99_in_a(X9))
U31_ga(T265, T9, qs99_out_a(X9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga(.(T265, []), T9) → U32_ga(T265, T9, qs99_in_a(T268))
U32_ga(T265, T9, qs99_out_a(T268)) → U33_ga(T265, T9, qs99_in_a(X10))
U33_ga(T265, T9, qs99_out_a(X10)) → qs1_out_ga(.(T265, []), T9)
U32_ga(T265, T9, qs99_out_a(T268)) → U34_ga(T265, T9, T268, qs99_in_a(T273))
U34_ga(T265, T9, T268, qs99_out_a(T273)) → U35_ga(T265, T9, app54_in_ggga(T268, T265, T273, T9))
app54_in_ggga(.(T210, T211), T212, T213, .(T210, T215)) → U15_ggga(T210, T211, T212, T213, T215, app54_in_ggga(T211, T212, T213, T215))
app54_in_ggga([], T225, T226, .(T225, T226)) → app54_out_ggga([], T225, T226, .(T225, T226))
U15_ggga(T210, T211, T212, T213, T215, app54_out_ggga(T211, T212, T213, T215)) → app54_out_ggga(.(T210, T211), T212, T213, .(T210, T215))
U35_ga(T265, T9, app54_out_ggga(T268, T265, T273, T9)) → qs1_out_ga(.(T265, []), T9)
qs1_in_ga([], []) → qs1_out_ga([], [])

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x3, x5)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
qs1_out_ga(x1, x2)  =  qs1_out_ga
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
part20_in_ggaa(x1, x2, x3, x4)  =  part20_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x3, x6)
part20_out_ggaa(x1, x2, x3, x4)  =  part20_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x3, x6)
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x6)
[]  =  []
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x5)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
U19_aa(x1, x2, x3, x4, x5)  =  U19_aa(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_aa(x1, x2)  =  qs1_out_aa
U20_aa(x1, x2, x3, x4, x5)  =  U20_aa(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(x3)
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(x6)
U4_gaaa(x1, x2, x3, x4, x5, x6)  =  U4_gaaa(x1, x6)
le33_in_ga(x1, x2)  =  le33_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
le33_out_ga(x1, x2)  =  le33_out_ga
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x6)
U21_aa(x1, x2, x3, x4, x5)  =  U21_aa(x1, x5)
U22_aa(x1, x2, x3, x4, x5)  =  U22_aa(x5)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
le33_in_aa(x1, x2)  =  le33_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
le33_out_aa(x1, x2)  =  le33_out_aa(x1)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x1, x5)
U28_aa(x1, x2, x3, x4, x5)  =  U28_aa(x5)
qs53_in_aa(x1, x2)  =  qs53_in_aa
U7_aa(x1, x2, x3, x4)  =  U7_aa(x4)
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)
qs53_out_aa(x1, x2)  =  qs53_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4)  =  U11_aa(x4)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
app65_in_aaaa(x1, x2, x3, x4)  =  app65_in_aaaa
U14_aaaa(x1, x2, x3, x4, x5, x6)  =  U14_aaaa(x6)
app65_out_aaaa(x1, x2, x3, x4)  =  app65_out_aaaa
U29_aa(x1, x2, x3, x4, x5, x6)  =  U29_aa(x1, x6)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
p52_in_aaaga(x1, x2, x3, x4, x5)  =  p52_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p52_out_aaaga(x1, x2, x3, x4, x5)  =  p52_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
app54_in_agaa(x1, x2, x3, x4)  =  app54_in_agaa(x2)
U15_agaa(x1, x2, x3, x4, x5, x6)  =  U15_agaa(x6)
app54_out_agaa(x1, x2, x3, x4)  =  app54_out_agaa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
qs99_in_a(x1)  =  qs99_in_a
qs99_out_a(x1)  =  qs99_out_a(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x3, x4)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
app54_in_gaga(x1, x2, x3, x4)  =  app54_in_gaga(x1, x3)
U15_gaga(x1, x2, x3, x4, x5, x6)  =  U15_gaga(x6)
app54_out_gaga(x1, x2, x3, x4)  =  app54_out_gaga
U23_aa(x1, x2, x3, x4, x5, x6)  =  U23_aa(x1, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x5)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x3, x5)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x5)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x5)
U29_ga(x1, x2, x3, x4, x5, x6)  =  U29_ga(x1, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x5)
U31_ga(x1, x2, x3)  =  U31_ga(x3)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x3)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x3, x4)
U35_ga(x1, x2, x3)  =  U35_ga(x3)
app54_in_ggga(x1, x2, x3, x4)  =  app54_in_ggga(x1, x2, x3)
U15_ggga(x1, x2, x3, x4, x5, x6)  =  U15_ggga(x1, x6)
app54_out_ggga(x1, x2, x3, x4)  =  app54_out_ggga(x4)
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x3, x6)

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

(257) UsableRulesProof (EQUIVALENT transformation)

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

(258) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
U2_GGAA(T61, T62, T63, X119, X120, gt10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U4_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
U4_GGAA(T83, T84, T85, X170, X171, le33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

gt10_in_gg(s(T37), s(T38)) → U1_gg(T37, T38, gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg(s(0), 0)
le33_in_gg(s(T98), s(T99)) → U6_gg(T98, T99, le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg(0, s(T106))
le33_in_gg(0, 0) → le33_out_gg(0, 0)
U1_gg(T37, T38, gt10_out_gg(T37, T38)) → gt10_out_gg(s(T37), s(T38))
U6_gg(T98, T99, le33_out_gg(T98, T99)) → le33_out_gg(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
gt10_in_gg(x1, x2)  =  gt10_in_gg(x1, x2)
s(x1)  =  s(x1)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
gt10_out_gg(x1, x2)  =  gt10_out_gg
le33_in_gg(x1, x2)  =  le33_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
le33_out_gg(x1, x2)  =  le33_out_gg
PART20_IN_GGAA(x1, x2, x3, x4)  =  PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x3, x6)

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

(259) PiDPToQDPProof (SOUND transformation)

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

(260) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63)) → U2_GGAA(T61, T63, gt10_in_gg(T61, T62))
U2_GGAA(T61, T63, gt10_out_gg) → PART20_IN_GGAA(T61, T63)
PART20_IN_GGAA(T83, .(T84, T85)) → U4_GGAA(T83, T85, le33_in_gg(T83, T84))
U4_GGAA(T83, T85, le33_out_gg) → PART20_IN_GGAA(T83, T85)

The TRS R consists of the following rules:

gt10_in_gg(s(T37), s(T38)) → U1_gg(gt10_in_gg(T37, T38))
gt10_in_gg(s(0), 0) → gt10_out_gg
le33_in_gg(s(T98), s(T99)) → U6_gg(le33_in_gg(T98, T99))
le33_in_gg(0, s(T106)) → le33_out_gg
le33_in_gg(0, 0) → le33_out_gg
U1_gg(gt10_out_gg) → gt10_out_gg
U6_gg(le33_out_gg) → le33_out_gg

The set Q consists of the following terms:

gt10_in_gg(x0, x1)
le33_in_gg(x0, x1)
U1_gg(x0)
U6_gg(x0)

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

(261) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• U2_GGAA(T61, T63, gt10_out_gg) → PART20_IN_GGAA(T61, T63)
  The graph contains the following edges 1 >= 1, 2 >= 2

• U4_GGAA(T83, T85, le33_out_gg) → PART20_IN_GGAA(T83, T85)
  The graph contains the following edges 1 >= 1, 2 >= 2

• PART20_IN_GGAA(T61, .(T62, T63)) → U2_GGAA(T61, T63, gt10_in_gg(T61, T62))
  The graph contains the following edges 1 >= 1, 2 > 2

• PART20_IN_GGAA(T83, .(T84, T85)) → U4_GGAA(T83, T85, le33_in_gg(T83, T84))
  The graph contains the following edges 1 >= 1, 2 > 2

(262) YES