Left Termination of the query pattern f_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

less(0, s(0)).
less(s(X), s(Y)) :- less(X, Y).
less(X, s(Y)) :- less(X, Y).
add(0, 0, 0).
add(s(X), Y, s(N)) :- add(X, Y, N).
add(X, s(Y), s(N)) :- add(X, Y, N).
f(X, N) :- ','(less(s(s(0)), X), add(N, s(0), X)).
f(X, N) :- ','(less(X, s(s(s(0)))), ','(add(X, s(s(0)), S), ','(f(S, N1), f(N1, N)))).

Queries:

f(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: f(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1, T2), SCOPE: 1, CLAUSE: 6 | f(T1, T2), SCOPE: 1, CLAUSE: 7\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(less(s(s(0)), T5), add(T7, s(0), T5)) | f(T5, T2), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 0 | ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 1 | ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 2 | ?_2 | f(T5, T2), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 1 | ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 2 | ?_2 | f(T5, T2), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 2 | ?_2 | f(T5, T2), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(less(s(0), T12), add(T7, s(0), s(T12)))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: less(s(0), T12)\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: add(T7, s(0), s(T12))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: less(s(0), T12), SCOPE: 3, CLAUSE: 0 | less(s(0), T12), SCOPE: 3, CLAUSE: 1 | less(s(0), T12), SCOPE: 3, CLAUSE: 2\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(s(0), T12), SCOPE: 3, CLAUSE: 1 | less(s(0), T12), SCOPE: 3, CLAUSE: 2\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: less(s(0), T12), SCOPE: 3, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(s(0), T12), SCOPE: 3, CLAUSE: 2\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(0, T19)\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(0, T19), SCOPE: 4, CLAUSE: 0 | less(0, T19), SCOPE: 4, CLAUSE: 1 | less(0, T19), SCOPE: 4, CLAUSE: 2\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: less(0, T19), SCOPE: 4, CLAUSE: 0\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: less(0, T19), SCOPE: 4, CLAUSE: 1 | less(0, T19), SCOPE: 4, CLAUSE: 2\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(0, T19), SCOPE: 4, CLAUSE: 2\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(0, T22)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(s(0), T25)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 3 | add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 4 | add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 5\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 4 | add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 5\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 4\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: add(T7, s(0), s(T12)), SCOPE: 5, CLAUSE: 5\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: add(T39, s(0), T38)\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: add(T39, s(0), T38), SCOPE: 6, CLAUSE: 3 | add(T39, s(0), T38), SCOPE: 6, CLAUSE: 4 | add(T39, s(0), T38), SCOPE: 6, CLAUSE: 5\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: add(T39, s(0), T38), SCOPE: 6, CLAUSE: 4 | add(T39, s(0), T38), SCOPE: 6, CLAUSE: 5\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: add(T39, s(0), T38), SCOPE: 6, CLAUSE: 4\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: add(T39, s(0), T38), SCOPE: 6, CLAUSE: 5\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: add(T52, s(0), T51)\n\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: add(T61, 0, T60)\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: add(T61, 0, T60), SCOPE: 7, CLAUSE: 3 | add(T61, 0, T60), SCOPE: 7, CLAUSE: 4 | add(T61, 0, T60), SCOPE: 7, CLAUSE: 5\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: add(T61, 0, T60), SCOPE: 7, CLAUSE: 3\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: add(T61, 0, T60), SCOPE: 7, CLAUSE: 4 | add(T61, 0, T60), SCOPE: 7, CLAUSE: 5\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: add(T61, 0, T60), SCOPE: 7, CLAUSE: 4\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: add(T61, 0, T60), SCOPE: 7, CLAUSE: 5\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: add(T72, 0, T71)\n\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: add(T81, 0, T80)\n\nT80 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(less(s(s(0)), T5), add(T7, s(0), T5)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ?_2 | f(T5, T2), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(less(s(s(0)), T86), add(T7, s(0), s(T86)))\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(s(s(0)), T86)\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: add(T7, s(0), s(T86))\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(s(s(0)), T86), SCOPE: 8, CLAUSE: 0 | less(s(s(0)), T86), SCOPE: 8, CLAUSE: 1 | less(s(s(0)), T86), SCOPE: 8, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: less(s(s(0)), T86), SCOPE: 8, CLAUSE: 1 | less(s(s(0)), T86), SCOPE: 8, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(s(s(0)), T86), SCOPE: 8, CLAUSE: 1\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(s(s(0)), T86), SCOPE: 8, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: less(s(0), T93)\n\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(s(s(0)), T97)\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: f(T5, T2), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(less(T107, s(s(s(0)))), ','(add(T107, s(s(0)), X193), ','(f(X193, X194), f(X194, T109))))\n\nT107 is ground\nX193 is free; \nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T107, s(s(s(0))))\n\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(add(T107, s(s(0)), X193), ','(f(X193, X194), f(X194, T109)))\n\nT107 is ground\nX193 is free; \nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 0 | less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 1 | less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 2\n\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 1 | less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 2\n\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 1\n\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: less(T107, s(s(s(0)))), SCOPE: 9, CLAUSE: 2\n\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: less(T116, s(s(0)))\n\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: less(T116, s(s(0))), SCOPE: 10, CLAUSE: 0 | less(T116, s(s(0))), SCOPE: 10, CLAUSE: 1 | less(T116, s(s(0))), SCOPE: 10, CLAUSE: 2\n\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: less(T116, s(s(0))), SCOPE: 10, CLAUSE: 1 | less(T116, s(s(0))), SCOPE: 10, CLAUSE: 2\n\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: less(T116, s(s(0))), SCOPE: 10, CLAUSE: 1\n\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: less(T116, s(s(0))), SCOPE: 10, CLAUSE: 2\n\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: less(T122, s(0))\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: less(T122, s(0)), SCOPE: 11, CLAUSE: 0 | less(T122, s(0)), SCOPE: 11, CLAUSE: 1 | less(T122, s(0)), SCOPE: 11, CLAUSE: 2\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: less(T122, s(0)), SCOPE: 11, CLAUSE: 0\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: less(T122, s(0)), SCOPE: 11, CLAUSE: 1 | less(T122, s(0)), SCOPE: 11, CLAUSE: 2\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: less(T122, s(0)), SCOPE: 11, CLAUSE: 1\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: less(T122, s(0)), SCOPE: 11, CLAUSE: 2\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: less(T127, 0)\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: less(T127, 0), SCOPE: 12, CLAUSE: 0 | less(T127, 0), SCOPE: 12, CLAUSE: 1 | less(T127, 0), SCOPE: 12, CLAUSE: 2\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: less(T127, 0), SCOPE: 12, CLAUSE: 1 | less(T127, 0), SCOPE: 12, CLAUSE: 2\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: less(T127, 0), SCOPE: 12, CLAUSE: 2\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: less(T136, 0)\n\nT136 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: less(T141, s(0))\n\nT141 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: less(T145, s(s(0)))\n\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: add(T107, s(s(0)), X193)\n\nT107 is ground\nX193 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(f(T148, X194), f(X194, T109))\n\nT148 is ground\nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 3 | add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 4 | add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 5\n\nT107 is ground\nX193 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 4 | add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 5\n\nT107 is ground\nX193 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 4\n\nT107 is ground\nX193 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: add(T107, s(s(0)), X193), SCOPE: 13, CLAUSE: 5\n\nT107 is ground\nX193 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: add(T154, s(s(0)), X302)\n\nT154 is ground\nX302 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: add(T159, s(0), X325)\n\nT159 is ground\nX325 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: add(T159, s(0), X325), SCOPE: 14, CLAUSE: 3 | add(T159, s(0), X325), SCOPE: 14, CLAUSE: 4 | add(T159, s(0), X325), SCOPE: 14, CLAUSE: 5\n\nT159 is ground\nX325 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: add(T159, s(0), X325), SCOPE: 14, CLAUSE: 4 | add(T159, s(0), X325), SCOPE: 14, CLAUSE: 5\n\nT159 is ground\nX325 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: add(T159, s(0), X325), SCOPE: 14, CLAUSE: 4\n\nT159 is ground\nX325 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: add(T159, s(0), X325), SCOPE: 14, CLAUSE: 5\n\nT159 is ground\nX325 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: add(T165, s(0), X349)\n\nT165 is ground\nX349 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: add(T169, 0, X368)\n\nT169 is ground\nX368 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: add(T169, 0, X368), SCOPE: 15, CLAUSE: 3 | add(T169, 0, X368), SCOPE: 15, CLAUSE: 4 | add(T169, 0, X368), SCOPE: 15, CLAUSE: 5\n\nT169 is ground\nX368 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: add(T169, 0, X368), SCOPE: 15, CLAUSE: 3\n\nT169 is ground\nX368 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: add(T169, 0, X368), SCOPE: 15, CLAUSE: 4 | add(T169, 0, X368), SCOPE: 15, CLAUSE: 5\n\nT169 is ground\nX368 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: add(T169, 0, X368), SCOPE: 15, CLAUSE: 4\n\nT169 is ground\nX368 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: add(T169, 0, X368), SCOPE: 15, CLAUSE: 5\n\nT169 is ground\nX368 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: add(T174, 0, X388)\n\nT174 is ground\nX388 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: f(T148, X194)\n\nT148 is ground\nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: f(T177, T109)\n\nT177 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: f(T148, X194), SCOPE: 16, CLAUSE: 6 | f(T148, X194), SCOPE: 16, CLAUSE: 7\n\nT148 is ground\nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: f(T148, X194), SCOPE: 16, CLAUSE: 6\n\nT148 is ground\nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: f(T148, X194), SCOPE: 16, CLAUSE: 7\n\nT148 is ground\nX194 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: ','(less(s(s(0)), T185), add(X419, s(0), T185))\n\nT185 is ground\nX419 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: less(s(s(0)), T185)\n\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: add(X419, s(0), T185)\n\nT185 is ground\nX419 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: add(X419, s(0), T185), SCOPE: 17, CLAUSE: 3 | add(X419, s(0), T185), SCOPE: 17, CLAUSE: 4 | add(X419, s(0), T185), SCOPE: 17, CLAUSE: 5\n\nT185 is ground\nX419 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: add(X419, s(0), T185), SCOPE: 17, CLAUSE: 4 | add(X419, s(0), T185), SCOPE: 17, CLAUSE: 5\n\nT185 is ground\nX419 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: add(X419, s(0), T185), SCOPE: 17, CLAUSE: 4\n\nT185 is ground\nX419 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: add(X419, s(0), T185), SCOPE: 17, CLAUSE: 5\n\nT185 is ground\nX419 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: add(X447, s(0), T193)\n\nT193 is ground\nX447 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: add(X466, 0, T197)\n\nT197 is ground\nX466 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: add(X466, 0, T197), SCOPE: 18, CLAUSE: 3 | add(X466, 0, T197), SCOPE: 18, CLAUSE: 4 | add(X466, 0, T197), SCOPE: 18, CLAUSE: 5\n\nT197 is ground\nX466 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: add(X466, 0, T197), SCOPE: 18, CLAUSE: 3\n\nT197 is ground\nX466 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: add(X466, 0, T197), SCOPE: 18, CLAUSE: 4 | add(X466, 0, T197), SCOPE: 18, CLAUSE: 5\n\nT197 is ground\nX466 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: add(X466, 0, T197), SCOPE: 18, CLAUSE: 4\n\nT197 is ground\nX466 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: add(X466, 0, T197), SCOPE: 18, CLAUSE: 5\n\nT197 is ground\nX466 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: add(X489, 0, T202)\n\nT202 is ground\nX489 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: ','(less(T206, s(s(s(0)))), ','(add(T206, s(s(0)), X509), ','(f(X509, X510), f(X510, X511))))\n\nT206 is ground\nX511 is free; \nX509 is free; \nX510 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: less(T206, s(s(s(0))))\n\nT206 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: ','(add(T206, s(s(0)), X509), ','(f(X509, X510), f(X510, X511)))\n\nT206 is ground\nX511 is free; \nX509 is free; \nX510 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: add(T206, s(s(0)), X509)\n\nT206 is ground\nX509 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: ','(f(T210, X510), f(X510, X511))\n\nT210 is ground\nX511 is free; \nX510 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: f(T210, X510)\n\nT210 is ground\nX510 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: f(T213, X511)\n\nT213 is ground\nX511 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 163 [arrowhead = none ];
163 -> 2;

164 [label="ONLY EVAL with clause\nf(X3, X4) :- ','(less(s(s(0)), X3), add(X4, s(0), X3)).\nand substitution\nT1 -> T5\nX3 -> T5\nT2 -> T7\nX4 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = white];
2 -> 164 [arrowhead = none ];
164 -> 3;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 165 [arrowhead = none ];
165 -> 4;

166 [label="BACKTRACK\nfor clause: less(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
4 -> 166 [arrowhead = none ];
166 -> 5;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 167 [arrowhead = none ];
167 -> 6;

168 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 168 [arrowhead = none ];
168 -> 7;

169 [label="EVAL with clause\nless(s(X13), s(X14)) :- less(X13, X14).\nand substitution\nX13 -> s(0)\nX14 -> T12\nT5 -> s(T12)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 169 [arrowhead = none ];
169 -> 8;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 170 [arrowhead = none ];
170 -> 9;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 171 [arrowhead = none ];
171 -> 55;

172 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 172 [arrowhead = none ];
172 -> 56;

173 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 173 [arrowhead = none ];
173 -> 10;

174 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
8 -> 174 [arrowhead = none ];
174 -> 11;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 175 [arrowhead = none ];
175 -> 12;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 176 [arrowhead = none ];
176 -> 29;

177 [label="BACKTRACK\nfor clause: less(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 177 [arrowhead = none ];
177 -> 13;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 178 [arrowhead = none ];
178 -> 14;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 179 [arrowhead = none ];
179 -> 15;

180 [label="EVAL with clause\nless(s(X29), s(X30)) :- less(X29, X30).\nand substitution\nX29 -> 0\nX30 -> T19\nT12 -> s(T19)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 180 [arrowhead = none ];
180 -> 16;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 181 [arrowhead = none ];
181 -> 17;

182 [label="EVAL with clause\nless(X47, s(X48)) :- less(X47, X48).\nand substitution\nX47 -> s(0)\nX48 -> T25\nT12 -> s(T25)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 182 [arrowhead = none ];
182 -> 27;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 183 [arrowhead = none ];
183 -> 28;

184 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 184 [arrowhead = none ];
184 -> 18;

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 185 [arrowhead = none ];
185 -> 19;

186 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 186 [arrowhead = none ];
186 -> 20;

187 [label="EVAL with clause\nless(0, s(0)).\nand substitution\nT19 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 187 [arrowhead = none ];
187 -> 21;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 188 [arrowhead = none ];
188 -> 22;

189 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
20 -> 189 [arrowhead = none ];
189 -> 24;

190 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 190 [arrowhead = none ];
190 -> 23;

191 [label="EVAL with clause\nless(X39, s(X40)) :- less(X39, X40).\nand substitution\nX39 -> 0\nX40 -> T22\nT19 -> s(T22)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 191 [arrowhead = none ];
191 -> 25;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 192 [arrowhead = none ];
192 -> 26;

193 [label="INSTANCE with matching:\nT19 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 193 [arrowhead = none , style=dashed];
193 -> 16[style=dashed];

194 [label="INSTANCE with matching:\nT12 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 194 [arrowhead = none , style=dashed];
194 -> 10[style=dashed];

195 [label="BACKTRACK\nfor clause: add(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
29 -> 195 [arrowhead = none ];
195 -> 30;

196 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 196 [arrowhead = none ];
196 -> 31;

197 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 197 [arrowhead = none ];
197 -> 32;

198 [label="EVAL with clause\nadd(s(X68), X69, s(X70)) :- add(X68, X69, X70).\nand substitution\nX68 -> T39\nT7 -> s(T39)\nX69 -> s(0)\nT12 -> T38\nX70 -> T38\nT37 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 198 [arrowhead = none ];
198 -> 33;

199 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 199 [arrowhead = none ];
199 -> 34;

200 [label="ONLY EVAL with clause\nadd(X137, s(X138), s(X139)) :- add(X137, X138, X139).\nand substitution\nT7 -> T81\nX137 -> T81\nX138 -> 0\nT12 -> T80\nX139 -> T80\nT79 -> T81", fontsize=14, style = filled, fillcolor = white];
32 -> 200 [arrowhead = none ];
200 -> 54;

201 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 201 [arrowhead = none ];
201 -> 35;

202 [label="BACKTRACK\nfor clause: add(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
35 -> 202 [arrowhead = none ];
202 -> 36;

203 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 203 [arrowhead = none ];
203 -> 37;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 204 [arrowhead = none ];
204 -> 38;

205 [label="EVAL with clause\nadd(s(X89), X90, s(X91)) :- add(X89, X90, X91).\nand substitution\nX89 -> T52\nT39 -> s(T52)\nX90 -> s(0)\nX91 -> T51\nT38 -> s(T51)\nT50 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 205 [arrowhead = none ];
205 -> 39;

206 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 206 [arrowhead = none ];
206 -> 40;

207 [label="EVAL with clause\nadd(X104, s(X105), s(X106)) :- add(X104, X105, X106).\nand substitution\nT39 -> T61\nX104 -> T61\nX105 -> 0\nX106 -> T60\nT38 -> s(T60)\nT59 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 207 [arrowhead = none ];
207 -> 41;

208 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 208 [arrowhead = none ];
208 -> 42;

209 [label="INSTANCE with matching:\nT39 -> T52\nT38 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 209 [arrowhead = none , style=dashed];
209 -> 33[style=dashed];

210 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 210 [arrowhead = none ];
210 -> 43;

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 211 [arrowhead = none ];
211 -> 44;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 212 [arrowhead = none ];
212 -> 45;

213 [label="EVAL with clause\nadd(0, 0, 0).\nand substitution\nT61 -> 0\nT60 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 213 [arrowhead = none ];
213 -> 46;

214 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 214 [arrowhead = none ];
214 -> 47;

215 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 215 [arrowhead = none ];
215 -> 49;

216 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 216 [arrowhead = none ];
216 -> 50;

217 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 217 [arrowhead = none ];
217 -> 48;

218 [label="EVAL with clause\nadd(s(X122), X123, s(X124)) :- add(X122, X123, X124).\nand substitution\nX122 -> T72\nT61 -> s(T72)\nX123 -> 0\nX124 -> T71\nT60 -> s(T71)\nT70 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 218 [arrowhead = none ];
218 -> 51;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 219 [arrowhead = none ];
219 -> 52;

220 [label="BACKTRACK\nfor clause: add(X, s(Y), s(N)) :- add(X, Y, N)because of non-unification", fontsize=14, style = filled, fillcolor = white];
50 -> 220 [arrowhead = none ];
220 -> 53;

221 [label="INSTANCE with matching:\nT61 -> T72\nT60 -> T71", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 221 [arrowhead = none , style=dashed];
221 -> 41[style=dashed];

222 [label="INSTANCE with matching:\nT61 -> T81\nT60 -> T80", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 222 [arrowhead = none , style=dashed];
222 -> 41[style=dashed];

223 [label="EVAL with clause\nless(X148, s(X149)) :- less(X148, X149).\nand substitution\nX148 -> s(s(0))\nX149 -> T86\nT5 -> s(T86)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 223 [arrowhead = none ];
223 -> 57;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 224 [arrowhead = none ];
224 -> 58;

225 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 225 [arrowhead = none ];
225 -> 69;

226 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 226 [arrowhead = none ];
226 -> 59;

227 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
57 -> 227 [arrowhead = none ];
227 -> 60;

228 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 228 [arrowhead = none ];
228 -> 61;

229 [label="INSTANCE with matching:\nT12 -> T86", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 229 [arrowhead = none , style=dashed];
229 -> 11[style=dashed];

230 [label="BACKTRACK\nfor clause: less(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
61 -> 230 [arrowhead = none ];
230 -> 62;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 231 [arrowhead = none ];
231 -> 63;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 232 [arrowhead = none ];
232 -> 64;

233 [label="EVAL with clause\nless(s(X164), s(X165)) :- less(X164, X165).\nand substitution\nX164 -> s(0)\nX165 -> T93\nT86 -> s(T93)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 233 [arrowhead = none ];
233 -> 65;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 234 [arrowhead = none ];
234 -> 66;

235 [label="EVAL with clause\nless(X174, s(X175)) :- less(X174, X175).\nand substitution\nX174 -> s(s(0))\nX175 -> T97\nT86 -> s(T97)", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 235 [arrowhead = none ];
235 -> 67;

236 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 236 [arrowhead = none ];
236 -> 68;

237 [label="INSTANCE with matching:\nT12 -> T93", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 237 [arrowhead = none , style=dashed];
237 -> 10[style=dashed];

238 [label="INSTANCE with matching:\nT86 -> T97", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 238 [arrowhead = none , style=dashed];
238 -> 59[style=dashed];

239 [label="ONLY EVAL with clause\nf(X191, X192) :- ','(less(X191, s(s(s(0)))), ','(add(X191, s(s(0)), X193), ','(f(X193, X194), f(X194, X192)))).\nand substitution\nT5 -> T107\nX191 -> T107\nT2 -> T109\nX192 -> T109\nT108 -> T109", fontsize=14, style = filled, fillcolor = white];
69 -> 239 [arrowhead = none ];
239 -> 70;

240 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
70 -> 240 [arrowhead = none ];
240 -> 71;

241 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
70 -> 241 [arrowhead = none ];
241 -> 72;

242 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 242 [arrowhead = none ];
242 -> 73;

243 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 243 [arrowhead = none ];
243 -> 102;

244 [label="SPLIT 2\nnew knowledge:\nT148 is ground\nreplacements:\nX193 -> T148", fontsize=14, style = filled, fillcolor = violet];
72 -> 244 [arrowhead = none ];
244 -> 103;

245 [label="BACKTRACK\nfor clause: less(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
73 -> 245 [arrowhead = none ];
245 -> 74;

246 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 246 [arrowhead = none ];
246 -> 75;

247 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 247 [arrowhead = none ];
247 -> 76;

248 [label="EVAL with clause\nless(s(X207), s(X208)) :- less(X207, X208).\nand substitution\nX207 -> T116\nT107 -> s(T116)\nX208 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 248 [arrowhead = none ];
248 -> 77;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 249 [arrowhead = none ];
249 -> 78;

250 [label="ONLY EVAL with clause\nless(X271, s(X272)) :- less(X271, X272).\nand substitution\nT107 -> T145\nX271 -> T145\nX272 -> s(s(0))", fontsize=14, style = filled, fillcolor = white];
76 -> 250 [arrowhead = none ];
250 -> 101;

251 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 251 [arrowhead = none ];
251 -> 79;

252 [label="BACKTRACK\nfor clause: less(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
79 -> 252 [arrowhead = none ];
252 -> 80;

253 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
80 -> 253 [arrowhead = none ];
253 -> 81;

254 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
80 -> 254 [arrowhead = none ];
254 -> 82;

255 [label="EVAL with clause\nless(s(X221), s(X222)) :- less(X221, X222).\nand substitution\nX221 -> T122\nT116 -> s(T122)\nX222 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 255 [arrowhead = none ];
255 -> 83;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 256 [arrowhead = none ];
256 -> 84;

257 [label="ONLY EVAL with clause\nless(X263, s(X264)) :- less(X263, X264).\nand substitution\nT116 -> T141\nX263 -> T141\nX264 -> s(0)", fontsize=14, style = filled, fillcolor = white];
82 -> 257 [arrowhead = none ];
257 -> 100;

258 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 258 [arrowhead = none ];
258 -> 85;

259 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 259 [arrowhead = none ];
259 -> 86;

260 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 260 [arrowhead = none ];
260 -> 87;

261 [label="EVAL with clause\nless(0, s(0)).\nand substitution\nT122 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 261 [arrowhead = none ];
261 -> 88;

262 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 262 [arrowhead = none ];
262 -> 89;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 263 [arrowhead = none ];
263 -> 91;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 264 [arrowhead = none ];
264 -> 92;

265 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 265 [arrowhead = none ];
265 -> 90;

266 [label="EVAL with clause\nless(s(X233), s(X234)) :- less(X233, X234).\nand substitution\nX233 -> T127\nT122 -> s(T127)\nX234 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 266 [arrowhead = none ];
266 -> 93;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 267 [arrowhead = none ];
267 -> 94;

268 [label="ONLY EVAL with clause\nless(X253, s(X254)) :- less(X253, X254).\nand substitution\nT122 -> T136\nX253 -> T136\nX254 -> 0", fontsize=14, style = filled, fillcolor = white];
92 -> 268 [arrowhead = none ];
268 -> 99;

269 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
93 -> 269 [arrowhead = none ];
269 -> 95;

270 [label="BACKTRACK\nfor clause: less(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
95 -> 270 [arrowhead = none ];
270 -> 96;

271 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
96 -> 271 [arrowhead = none ];
271 -> 97;

272 [label="BACKTRACK\nfor clause: less(X, s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
97 -> 272 [arrowhead = none ];
272 -> 98;

273 [label="INSTANCE with matching:\nT127 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
99 -> 273 [arrowhead = none , style=dashed];
273 -> 93[style=dashed];

274 [label="INSTANCE with matching:\nT122 -> T141", fontsize=14, style = filled, fillcolor = lightgrey];
100 -> 274 [arrowhead = none , style=dashed];
274 -> 83[style=dashed];

275 [label="INSTANCE with matching:\nT116 -> T145", fontsize=14, style = filled, fillcolor = lightgrey];
101 -> 275 [arrowhead = none , style=dashed];
275 -> 77[style=dashed];

276 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 276 [arrowhead = none ];
276 -> 104;

277 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
103 -> 277 [arrowhead = none ];
277 -> 129;

278 [label="SPLIT 2\nnew knowledge:\nT177 is ground\nreplacements:\nX194 -> T177", fontsize=14, style = filled, fillcolor = violet];
103 -> 278 [arrowhead = none ];
278 -> 130;

279 [label="BACKTRACK\nfor clause: add(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
104 -> 279 [arrowhead = none ];
279 -> 105;

280 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
105 -> 280 [arrowhead = none ];
280 -> 106;

281 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
105 -> 281 [arrowhead = none ];
281 -> 107;

282 [label="EVAL with clause\nadd(s(X299), X300, s(X301)) :- add(X299, X300, X301).\nand substitution\nX299 -> T154\nT107 -> s(T154)\nX300 -> s(s(0))\nX301 -> X302\nX193 -> s(X302)", fontsize=14, style = filled, fillcolor = lightblue];
106 -> 282 [arrowhead = none ];
282 -> 108;

283 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
106 -> 283 [arrowhead = none ];
283 -> 109;

284 [label="ONLY EVAL with clause\nadd(X322, s(X323), s(X324)) :- add(X322, X323, X324).\nand substitution\nT107 -> T159\nX322 -> T159\nX323 -> s(0)\nX324 -> X325\nX193 -> s(X325)", fontsize=14, style = filled, fillcolor = white];
107 -> 284 [arrowhead = none ];
284 -> 110;

285 [label="INSTANCE with matching:\nT107 -> T154\nX193 -> X302", fontsize=14, style = filled, fillcolor = lightgrey];
108 -> 285 [arrowhead = none , style=dashed];
285 -> 102[style=dashed];

286 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
110 -> 286 [arrowhead = none ];
286 -> 111;

287 [label="BACKTRACK\nfor clause: add(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
111 -> 287 [arrowhead = none ];
287 -> 112;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
112 -> 288 [arrowhead = none ];
288 -> 113;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
112 -> 289 [arrowhead = none ];
289 -> 114;

290 [label="EVAL with clause\nadd(s(X346), X347, s(X348)) :- add(X346, X347, X348).\nand substitution\nX346 -> T165\nT159 -> s(T165)\nX347 -> s(0)\nX348 -> X349\nX325 -> s(X349)", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 290 [arrowhead = none ];
290 -> 115;

291 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 291 [arrowhead = none ];
291 -> 116;

292 [label="ONLY EVAL with clause\nadd(X365, s(X366), s(X367)) :- add(X365, X366, X367).\nand substitution\nT159 -> T169\nX365 -> T169\nX366 -> 0\nX367 -> X368\nX325 -> s(X368)", fontsize=14, style = filled, fillcolor = white];
114 -> 292 [arrowhead = none ];
292 -> 117;

293 [label="INSTANCE with matching:\nT159 -> T165\nX325 -> X349", fontsize=14, style = filled, fillcolor = lightgrey];
115 -> 293 [arrowhead = none , style=dashed];
293 -> 110[style=dashed];

294 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
117 -> 294 [arrowhead = none ];
294 -> 118;

295 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
118 -> 295 [arrowhead = none ];
295 -> 119;

296 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
118 -> 296 [arrowhead = none ];
296 -> 120;

297 [label="EVAL with clause\nadd(0, 0, 0).\nand substitution\nT169 -> 0\nX368 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
119 -> 297 [arrowhead = none ];
297 -> 121;

298 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
119 -> 298 [arrowhead = none ];
298 -> 122;

299 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
120 -> 299 [arrowhead = none ];
299 -> 124;

300 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
120 -> 300 [arrowhead = none ];
300 -> 125;

301 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
121 -> 301 [arrowhead = none ];
301 -> 123;

302 [label="EVAL with clause\nadd(s(X385), X386, s(X387)) :- add(X385, X386, X387).\nand substitution\nX385 -> T174\nT169 -> s(T174)\nX386 -> 0\nX387 -> X388\nX368 -> s(X388)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 302 [arrowhead = none ];
302 -> 126;

303 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 303 [arrowhead = none ];
303 -> 127;

304 [label="BACKTRACK\nfor clause: add(X, s(Y), s(N)) :- add(X, Y, N)because of non-unification", fontsize=14, style = filled, fillcolor = white];
125 -> 304 [arrowhead = none ];
304 -> 128;

305 [label="INSTANCE with matching:\nT169 -> T174\nX368 -> X388", fontsize=14, style = filled, fillcolor = lightgrey];
126 -> 305 [arrowhead = none , style=dashed];
305 -> 117[style=dashed];

306 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
129 -> 306 [arrowhead = none ];
306 -> 131;

307 [label="INSTANCE with matching:\nT1 -> T177\nT2 -> T109", fontsize=14, style = filled, fillcolor = lightgrey];
130 -> 307 [arrowhead = none , style=dashed];
307 -> 1[style=dashed];

308 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
131 -> 308 [arrowhead = none ];
308 -> 132;

309 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
131 -> 309 [arrowhead = none ];
309 -> 133;

310 [label="ONLY EVAL with clause\nf(X417, X418) :- ','(less(s(s(0)), X417), add(X418, s(0), X417)).\nand substitution\nT148 -> T185\nX417 -> T185\nX194 -> X419\nX418 -> X419", fontsize=14, style = filled, fillcolor = white];
132 -> 310 [arrowhead = none ];
310 -> 134;

311 [label="ONLY EVAL with clause\nf(X507, X508) :- ','(less(X507, s(s(s(0)))), ','(add(X507, s(s(0)), X509), ','(f(X509, X510), f(X510, X508)))).\nand substitution\nT148 -> T206\nX507 -> T206\nX194 -> X511\nX508 -> X511", fontsize=14, style = filled, fillcolor = white];
133 -> 311 [arrowhead = none ];
311 -> 156;

312 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
134 -> 312 [arrowhead = none ];
312 -> 135;

313 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
134 -> 313 [arrowhead = none ];
313 -> 136;

314 [label="INSTANCE with matching:\nT86 -> T185", fontsize=14, style = filled, fillcolor = lightgrey];
135 -> 314 [arrowhead = none , style=dashed];
314 -> 59[style=dashed];

315 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
136 -> 315 [arrowhead = none ];
315 -> 137;

316 [label="BACKTRACK\nfor clause: add(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
137 -> 316 [arrowhead = none ];
316 -> 138;

317 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
138 -> 317 [arrowhead = none ];
317 -> 139;

318 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
138 -> 318 [arrowhead = none ];
318 -> 140;

319 [label="EVAL with clause\nadd(s(X444), X445, s(X446)) :- add(X444, X445, X446).\nand substitution\nX444 -> X447\nX419 -> s(X447)\nX445 -> s(0)\nX446 -> T193\nT185 -> s(T193)", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 319 [arrowhead = none ];
319 -> 141;

320 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 320 [arrowhead = none ];
320 -> 142;

321 [label="EVAL with clause\nadd(X463, s(X464), s(X465)) :- add(X463, X464, X465).\nand substitution\nX419 -> X466\nX463 -> X466\nX464 -> 0\nX465 -> T197\nT185 -> s(T197)", fontsize=14, style = filled, fillcolor = lightblue];
140 -> 321 [arrowhead = none ];
321 -> 143;

322 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
140 -> 322 [arrowhead = none ];
322 -> 144;

323 [label="INSTANCE with matching:\nX419 -> X447\nT185 -> T193", fontsize=14, style = filled, fillcolor = lightgrey];
141 -> 323 [arrowhead = none , style=dashed];
323 -> 136[style=dashed];

324 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
143 -> 324 [arrowhead = none ];
324 -> 145;

325 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
145 -> 325 [arrowhead = none ];
325 -> 146;

326 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
145 -> 326 [arrowhead = none ];
326 -> 147;

327 [label="EVAL with clause\nadd(0, 0, 0).\nand substitution\nX466 -> 0\nT197 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
146 -> 327 [arrowhead = none ];
327 -> 148;

328 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
146 -> 328 [arrowhead = none ];
328 -> 149;

329 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
147 -> 329 [arrowhead = none ];
329 -> 151;

330 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
147 -> 330 [arrowhead = none ];
330 -> 152;

331 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
148 -> 331 [arrowhead = none ];
331 -> 150;

332 [label="EVAL with clause\nadd(s(X486), X487, s(X488)) :- add(X486, X487, X488).\nand substitution\nX486 -> X489\nX466 -> s(X489)\nX487 -> 0\nX488 -> T202\nT197 -> s(T202)", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 332 [arrowhead = none ];
332 -> 153;

333 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 333 [arrowhead = none ];
333 -> 154;

334 [label="BACKTRACK\nfor clause: add(X, s(Y), s(N)) :- add(X, Y, N)because of non-unification", fontsize=14, style = filled, fillcolor = white];
152 -> 334 [arrowhead = none ];
334 -> 155;

335 [label="INSTANCE with matching:\nX466 -> X489\nT197 -> T202", fontsize=14, style = filled, fillcolor = lightgrey];
153 -> 335 [arrowhead = none , style=dashed];
335 -> 143[style=dashed];

336 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
156 -> 336 [arrowhead = none ];
336 -> 157;

337 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
156 -> 337 [arrowhead = none ];
337 -> 158;

338 [label="INSTANCE with matching:\nT107 -> T206", fontsize=14, style = filled, fillcolor = lightgrey];
157 -> 338 [arrowhead = none , style=dashed];
338 -> 71[style=dashed];

339 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
158 -> 339 [arrowhead = none ];
339 -> 159;

340 [label="SPLIT 2\nnew knowledge:\nT210 is ground\nreplacements:\nX509 -> T210", fontsize=14, style = filled, fillcolor = violet];
158 -> 340 [arrowhead = none ];
340 -> 160;

341 [label="INSTANCE with matching:\nT107 -> T206\nX193 -> X509", fontsize=14, style = filled, fillcolor = lightgrey];
159 -> 341 [arrowhead = none , style=dashed];
341 -> 102[style=dashed];

342 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
160 -> 342 [arrowhead = none ];
342 -> 161;

343 [label="SPLIT 2\nnew knowledge:\nT213 is ground\nreplacements:\nX510 -> T213", fontsize=14, style = filled, fillcolor = violet];
160 -> 343 [arrowhead = none ];
343 -> 162;

344 [label="INSTANCE with matching:\nT148 -> T210\nX194 -> X510", fontsize=14, style = filled, fillcolor = lightgrey];
161 -> 344 [arrowhead = none , style=dashed];
344 -> 129[style=dashed];

345 [label="INSTANCE with matching:\nT148 -> T213\nX194 -> X511", fontsize=14, style = filled, fillcolor = lightgrey];
162 -> 345 [arrowhead = none , style=dashed];
345 -> 129[style=dashed];

}

(2) Obligation:

Clauses:

less16(s(0)).
less16(s(T22)) :- less16(T22).
less10(s(T19)) :- less16(T19).
less10(s(T25)) :- less10(T25).
add33(s(T52), s(T51)) :- add33(T52, T51).
add33(T61, s(T60)) :- add41(T61, T60).
add41(0, 0).
add41(s(T72), s(T71)) :- add41(T72, T71).
add11(s(T39), T38) :- add33(T39, T38).
add11(T81, T80) :- add41(T81, T80).
less59(s(T93)) :- less10(T93).
less59(s(T97)) :- less59(T97).
less83(0).
less77(s(T122)) :- less83(T122).
less77(T141) :- less83(T141).
add102(s(T154), s(X302)) :- add102(T154, X302).
add102(T159, s(X325)) :- add110(T159, X325).
add110(s(T165), s(X349)) :- add110(T165, X349).
add110(T169, s(X368)) :- add117(T169, X368).
add117(0, 0).
add117(s(T174), s(X388)) :- add117(T174, X388).
add136(s(X447), s(T193)) :- add136(X447, T193).
add136(X466, s(T197)) :- add143(X466, T197).
add143(0, 0).
add143(s(X489), s(T202)) :- add143(X489, T202).
less71(s(T116)) :- less77(T116).
less71(T145) :- less77(T145).
f129(T185, X419) :- less59(T185).
f129(T185, X419) :- ','(less59(T185), add136(X419, T185)).
f129(T206, X511) :- less71(T206).
f129(T206, X511) :- ','(less71(T206), add102(T206, X509)).
f129(T206, X511) :- ','(less71(T206), ','(add102(T206, T210), f129(T210, X510))).
f129(T206, X511) :- ','(less71(T206), ','(add102(T206, T210), ','(f129(T210, T213), f129(T213, X511)))).
f1(s(T12), T7) :- less10(T12).
f1(s(T12), T7) :- ','(less10(T12), add11(T7, T12)).
f1(s(T86), T7) :- less59(T86).
f1(s(T86), T7) :- ','(less59(T86), add11(T7, T86)).
f1(T107, T109) :- less71(T107).
f1(T107, T109) :- ','(less71(T107), add102(T107, X193)).
f1(T107, T109) :- ','(less71(T107), ','(add102(T107, T148), f129(T148, X194))).
f1(T107, T109) :- ','(less71(T107), ','(add102(T107, T148), ','(f129(T148, T177), f1(T177, T109)))).

Queries:

f1(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:
f1_in: (b,f) (f,f)
less10_in: (b) (f)
less16_in: (b) (f)
add11_in: (f,b)
add33_in: (f,b)
add41_in: (f,b)
less59_in: (b) (f)
less71_in: (b) (f)
less77_in: (b) (f)
add102_in: (b,f)
add110_in: (b,f)
add117_in: (b,f)
f129_in: (b,f) (f,f)
add136_in: (f,b)
add143_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)

(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:

F1_IN_GA(s(T12), T7) → U31_GA(T12, T7, less10_in_g(T12))
F1_IN_GA(s(T12), T7) → LESS10_IN_G(T12)
LESS10_IN_G(s(T19)) → U2_G(T19, less16_in_g(T19))
LESS10_IN_G(s(T19)) → LESS16_IN_G(T19)
LESS16_IN_G(s(T22)) → U1_G(T22, less16_in_g(T22))
LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)
LESS10_IN_G(s(T25)) → U3_G(T25, less10_in_g(T25))
LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)
U31_GA(T12, T7, less10_out_g(T12)) → U32_GA(T12, T7, add11_in_ag(T7, T12))
U31_GA(T12, T7, less10_out_g(T12)) → ADD11_IN_AG(T7, T12)
ADD11_IN_AG(s(T39), T38) → U7_AG(T39, T38, add33_in_ag(T39, T38))
ADD11_IN_AG(s(T39), T38) → ADD33_IN_AG(T39, T38)
ADD33_IN_AG(s(T52), s(T51)) → U4_AG(T52, T51, add33_in_ag(T52, T51))
ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)
ADD33_IN_AG(T61, s(T60)) → U5_AG(T61, T60, add41_in_ag(T61, T60))
ADD33_IN_AG(T61, s(T60)) → ADD41_IN_AG(T61, T60)
ADD41_IN_AG(s(T72), s(T71)) → U6_AG(T72, T71, add41_in_ag(T72, T71))
ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)
ADD11_IN_AG(T81, T80) → U8_AG(T81, T80, add41_in_ag(T81, T80))
ADD11_IN_AG(T81, T80) → ADD41_IN_AG(T81, T80)
F1_IN_GA(s(T86), T7) → U33_GA(T86, T7, less59_in_g(T86))
F1_IN_GA(s(T86), T7) → LESS59_IN_G(T86)
LESS59_IN_G(s(T93)) → U9_G(T93, less10_in_g(T93))
LESS59_IN_G(s(T93)) → LESS10_IN_G(T93)
LESS59_IN_G(s(T97)) → U10_G(T97, less59_in_g(T97))
LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)
U33_GA(T86, T7, less59_out_g(T86)) → U34_GA(T86, T7, add11_in_ag(T7, T86))
U33_GA(T86, T7, less59_out_g(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_GA(T107, T109) → U35_GA(T107, T109, less71_in_g(T107))
F1_IN_GA(T107, T109) → LESS71_IN_G(T107)
LESS71_IN_G(s(T116)) → U21_G(T116, less77_in_g(T116))
LESS71_IN_G(s(T116)) → LESS77_IN_G(T116)
LESS77_IN_G(s(T122)) → U11_G(T122, less83_in_g(T122))
LESS77_IN_G(s(T122)) → LESS83_IN_G(T122)
LESS77_IN_G(T141) → U12_G(T141, less83_in_g(T141))
LESS77_IN_G(T141) → LESS83_IN_G(T141)
LESS71_IN_G(T145) → U22_G(T145, less77_in_g(T145))
LESS71_IN_G(T145) → LESS77_IN_G(T145)
U35_GA(T107, T109, less71_out_g(T107)) → U36_GA(T107, T109, add102_in_ga(T107, X193))
U35_GA(T107, T109, less71_out_g(T107)) → ADD102_IN_GA(T107, X193)
ADD102_IN_GA(s(T154), s(X302)) → U13_GA(T154, X302, add102_in_ga(T154, X302))
ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)
ADD102_IN_GA(T159, s(X325)) → U14_GA(T159, X325, add110_in_ga(T159, X325))
ADD102_IN_GA(T159, s(X325)) → ADD110_IN_GA(T159, X325)
ADD110_IN_GA(s(T165), s(X349)) → U15_GA(T165, X349, add110_in_ga(T165, X349))
ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)
ADD110_IN_GA(T169, s(X368)) → U16_GA(T169, X368, add117_in_ga(T169, X368))
ADD110_IN_GA(T169, s(X368)) → ADD117_IN_GA(T169, X368)
ADD117_IN_GA(s(T174), s(X388)) → U17_GA(T174, X388, add117_in_ga(T174, X388))
ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)
U35_GA(T107, T109, less71_out_g(T107)) → U37_GA(T107, T109, add102_in_ga(T107, T148))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U38_GA(T107, T109, f129_in_ga(T148, X194))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
F129_IN_GA(T185, X419) → U23_GA(T185, X419, less59_in_g(T185))
F129_IN_GA(T185, X419) → LESS59_IN_G(T185)
U23_GA(T185, X419, less59_out_g(T185)) → U24_GA(T185, X419, add136_in_ag(X419, T185))
U23_GA(T185, X419, less59_out_g(T185)) → ADD136_IN_AG(X419, T185)
ADD136_IN_AG(s(X447), s(T193)) → U18_AG(X447, T193, add136_in_ag(X447, T193))
ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)
ADD136_IN_AG(X466, s(T197)) → U19_AG(X466, T197, add143_in_ag(X466, T197))
ADD136_IN_AG(X466, s(T197)) → ADD143_IN_AG(X466, T197)
ADD143_IN_AG(s(X489), s(T202)) → U20_AG(X489, T202, add143_in_ag(X489, T202))
ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)
F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
F129_IN_GA(T206, X511) → LESS71_IN_G(T206)
U25_GA(T206, X511, less71_out_g(T206)) → U26_GA(T206, X511, add102_in_ga(T206, X509))
U25_GA(T206, X511, less71_out_g(T206)) → ADD102_IN_GA(T206, X509)
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U28_GA(T206, X511, f129_in_ga(T210, X510))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → U30_GA(T206, X511, f129_in_aa(T213, X511))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T185, X419) → U23_AA(T185, X419, less59_in_a(T185))
F129_IN_AA(T185, X419) → LESS59_IN_A(T185)
LESS59_IN_A(s(T93)) → U9_A(T93, less10_in_a(T93))
LESS59_IN_A(s(T93)) → LESS10_IN_A(T93)
LESS10_IN_A(s(T19)) → U2_A(T19, less16_in_a(T19))
LESS10_IN_A(s(T19)) → LESS16_IN_A(T19)
LESS16_IN_A(s(T22)) → U1_A(T22, less16_in_a(T22))
LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)
LESS10_IN_A(s(T25)) → U3_A(T25, less10_in_a(T25))
LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)
LESS59_IN_A(s(T97)) → U10_A(T97, less59_in_a(T97))
LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)
U23_AA(T185, X419, less59_out_a(T185)) → U24_AA(T185, X419, add136_in_ag(X419, T185))
U23_AA(T185, X419, less59_out_a(T185)) → ADD136_IN_AG(X419, T185)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
F129_IN_AA(T206, X511) → LESS71_IN_A(T206)
LESS71_IN_A(s(T116)) → U21_A(T116, less77_in_a(T116))
LESS71_IN_A(s(T116)) → LESS77_IN_A(T116)
LESS77_IN_A(s(T122)) → U11_A(T122, less83_in_a(T122))
LESS77_IN_A(s(T122)) → LESS83_IN_A(T122)
LESS77_IN_A(T141) → U12_A(T141, less83_in_a(T141))
LESS77_IN_A(T141) → LESS83_IN_A(T141)
LESS71_IN_A(T145) → U22_A(T145, less77_in_a(T145))
LESS71_IN_A(T145) → LESS77_IN_A(T145)
U25_AA(T206, X511, less71_out_a(T206)) → U26_AA(T206, X511, add102_in_ga(T206, X509))
U25_AA(T206, X511, less71_out_a(T206)) → ADD102_IN_GA(T206, X509)
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U28_AA(T206, X511, f129_in_ga(T210, X510))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → U30_AA(T206, X511, f129_in_aa(T213, X511))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U39_GA(T107, T109, f129_in_ga(T148, T177))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → U40_GA(T107, T109, f1_in_aa(T177, T109))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(s(T12), T7) → U31_AA(T12, T7, less10_in_a(T12))
F1_IN_AA(s(T12), T7) → LESS10_IN_A(T12)
U31_AA(T12, T7, less10_out_a(T12)) → U32_AA(T12, T7, add11_in_ag(T7, T12))
U31_AA(T12, T7, less10_out_a(T12)) → ADD11_IN_AG(T7, T12)
F1_IN_AA(s(T86), T7) → U33_AA(T86, T7, less59_in_a(T86))
F1_IN_AA(s(T86), T7) → LESS59_IN_A(T86)
U33_AA(T86, T7, less59_out_a(T86)) → U34_AA(T86, T7, add11_in_ag(T7, T86))
U33_AA(T86, T7, less59_out_a(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
F1_IN_AA(T107, T109) → LESS71_IN_A(T107)
U35_AA(T107, T109, less71_out_a(T107)) → U36_AA(T107, T109, add102_in_ga(T107, X193))
U35_AA(T107, T109, less71_out_a(T107)) → ADD102_IN_GA(T107, X193)
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U38_AA(T107, T109, f129_in_ga(T148, X194))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → U40_AA(T107, T109, f1_in_aa(T177, T109))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)
LESS16_IN_G(x1)  =  LESS16_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)
U32_GA(x1, x2, x3)  =  U32_GA(x3)
ADD11_IN_AG(x1, x2)  =  ADD11_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x3)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x3)
U8_AG(x1, x2, x3)  =  U8_AG(x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)
U9_G(x1, x2)  =  U9_G(x2)
U10_G(x1, x2)  =  U10_G(x2)
U34_GA(x1, x2, x3)  =  U34_GA(x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
LESS71_IN_G(x1)  =  LESS71_IN_G(x1)
U21_G(x1, x2)  =  U21_G(x2)
LESS77_IN_G(x1)  =  LESS77_IN_G(x1)
U11_G(x1, x2)  =  U11_G(x2)
LESS83_IN_G(x1)  =  LESS83_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x2)
U22_G(x1, x2)  =  U22_G(x2)
U36_GA(x1, x2, x3)  =  U36_GA(x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)
U13_GA(x1, x2, x3)  =  U13_GA(x3)
U14_GA(x1, x2, x3)  =  U14_GA(x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x3)
U16_GA(x1, x2, x3)  =  U16_GA(x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)
U17_GA(x1, x2, x3)  =  U17_GA(x3)
U37_GA(x1, x2, x3)  =  U37_GA(x3)
U38_GA(x1, x2, x3)  =  U38_GA(x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)
U18_AG(x1, x2, x3)  =  U18_AG(x3)
U19_AG(x1, x2, x3)  =  U19_AG(x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x3)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U26_GA(x1, x2, x3)  =  U26_GA(x3)
U27_GA(x1, x2, x3)  =  U27_GA(x3)
U28_GA(x1, x2, x3)  =  U28_GA(x3)
U29_GA(x1, x2, x3)  =  U29_GA(x3)
U30_GA(x1, x2, x3)  =  U30_GA(x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U23_AA(x1, x2, x3)  =  U23_AA(x3)
LESS59_IN_A(x1)  =  LESS59_IN_A
U9_A(x1, x2)  =  U9_A(x2)
LESS10_IN_A(x1)  =  LESS10_IN_A
U2_A(x1, x2)  =  U2_A(x2)
LESS16_IN_A(x1)  =  LESS16_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U10_A(x1, x2)  =  U10_A(x2)
U24_AA(x1, x2, x3)  =  U24_AA(x1, x3)
U25_AA(x1, x2, x3)  =  U25_AA(x3)
LESS71_IN_A(x1)  =  LESS71_IN_A
U21_A(x1, x2)  =  U21_A(x2)
LESS77_IN_A(x1)  =  LESS77_IN_A
U11_A(x1, x2)  =  U11_A(x2)
LESS83_IN_A(x1)  =  LESS83_IN_A
U12_A(x1, x2)  =  U12_A(x2)
U22_A(x1, x2)  =  U22_A(x2)
U26_AA(x1, x2, x3)  =  U26_AA(x1, x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U28_AA(x1, x2, x3)  =  U28_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)
U30_AA(x1, x2, x3)  =  U30_AA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x3)
U40_GA(x1, x2, x3)  =  U40_GA(x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U31_AA(x1, x2, x3)  =  U31_AA(x3)
U32_AA(x1, x2, x3)  =  U32_AA(x1, x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3)  =  U34_AA(x1, x3)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U36_AA(x1, x2, x3)  =  U36_AA(x1, x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U38_AA(x1, x2, x3)  =  U38_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)
U40_AA(x1, x2, x3)  =  U40_AA(x1, x3)

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

(6) Obligation:

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

F1_IN_GA(s(T12), T7) → U31_GA(T12, T7, less10_in_g(T12))
F1_IN_GA(s(T12), T7) → LESS10_IN_G(T12)
LESS10_IN_G(s(T19)) → U2_G(T19, less16_in_g(T19))
LESS10_IN_G(s(T19)) → LESS16_IN_G(T19)
LESS16_IN_G(s(T22)) → U1_G(T22, less16_in_g(T22))
LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)
LESS10_IN_G(s(T25)) → U3_G(T25, less10_in_g(T25))
LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)
U31_GA(T12, T7, less10_out_g(T12)) → U32_GA(T12, T7, add11_in_ag(T7, T12))
U31_GA(T12, T7, less10_out_g(T12)) → ADD11_IN_AG(T7, T12)
ADD11_IN_AG(s(T39), T38) → U7_AG(T39, T38, add33_in_ag(T39, T38))
ADD11_IN_AG(s(T39), T38) → ADD33_IN_AG(T39, T38)
ADD33_IN_AG(s(T52), s(T51)) → U4_AG(T52, T51, add33_in_ag(T52, T51))
ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)
ADD33_IN_AG(T61, s(T60)) → U5_AG(T61, T60, add41_in_ag(T61, T60))
ADD33_IN_AG(T61, s(T60)) → ADD41_IN_AG(T61, T60)
ADD41_IN_AG(s(T72), s(T71)) → U6_AG(T72, T71, add41_in_ag(T72, T71))
ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)
ADD11_IN_AG(T81, T80) → U8_AG(T81, T80, add41_in_ag(T81, T80))
ADD11_IN_AG(T81, T80) → ADD41_IN_AG(T81, T80)
F1_IN_GA(s(T86), T7) → U33_GA(T86, T7, less59_in_g(T86))
F1_IN_GA(s(T86), T7) → LESS59_IN_G(T86)
LESS59_IN_G(s(T93)) → U9_G(T93, less10_in_g(T93))
LESS59_IN_G(s(T93)) → LESS10_IN_G(T93)
LESS59_IN_G(s(T97)) → U10_G(T97, less59_in_g(T97))
LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)
U33_GA(T86, T7, less59_out_g(T86)) → U34_GA(T86, T7, add11_in_ag(T7, T86))
U33_GA(T86, T7, less59_out_g(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_GA(T107, T109) → U35_GA(T107, T109, less71_in_g(T107))
F1_IN_GA(T107, T109) → LESS71_IN_G(T107)
LESS71_IN_G(s(T116)) → U21_G(T116, less77_in_g(T116))
LESS71_IN_G(s(T116)) → LESS77_IN_G(T116)
LESS77_IN_G(s(T122)) → U11_G(T122, less83_in_g(T122))
LESS77_IN_G(s(T122)) → LESS83_IN_G(T122)
LESS77_IN_G(T141) → U12_G(T141, less83_in_g(T141))
LESS77_IN_G(T141) → LESS83_IN_G(T141)
LESS71_IN_G(T145) → U22_G(T145, less77_in_g(T145))
LESS71_IN_G(T145) → LESS77_IN_G(T145)
U35_GA(T107, T109, less71_out_g(T107)) → U36_GA(T107, T109, add102_in_ga(T107, X193))
U35_GA(T107, T109, less71_out_g(T107)) → ADD102_IN_GA(T107, X193)
ADD102_IN_GA(s(T154), s(X302)) → U13_GA(T154, X302, add102_in_ga(T154, X302))
ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)
ADD102_IN_GA(T159, s(X325)) → U14_GA(T159, X325, add110_in_ga(T159, X325))
ADD102_IN_GA(T159, s(X325)) → ADD110_IN_GA(T159, X325)
ADD110_IN_GA(s(T165), s(X349)) → U15_GA(T165, X349, add110_in_ga(T165, X349))
ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)
ADD110_IN_GA(T169, s(X368)) → U16_GA(T169, X368, add117_in_ga(T169, X368))
ADD110_IN_GA(T169, s(X368)) → ADD117_IN_GA(T169, X368)
ADD117_IN_GA(s(T174), s(X388)) → U17_GA(T174, X388, add117_in_ga(T174, X388))
ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)
U35_GA(T107, T109, less71_out_g(T107)) → U37_GA(T107, T109, add102_in_ga(T107, T148))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U38_GA(T107, T109, f129_in_ga(T148, X194))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
F129_IN_GA(T185, X419) → U23_GA(T185, X419, less59_in_g(T185))
F129_IN_GA(T185, X419) → LESS59_IN_G(T185)
U23_GA(T185, X419, less59_out_g(T185)) → U24_GA(T185, X419, add136_in_ag(X419, T185))
U23_GA(T185, X419, less59_out_g(T185)) → ADD136_IN_AG(X419, T185)
ADD136_IN_AG(s(X447), s(T193)) → U18_AG(X447, T193, add136_in_ag(X447, T193))
ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)
ADD136_IN_AG(X466, s(T197)) → U19_AG(X466, T197, add143_in_ag(X466, T197))
ADD136_IN_AG(X466, s(T197)) → ADD143_IN_AG(X466, T197)
ADD143_IN_AG(s(X489), s(T202)) → U20_AG(X489, T202, add143_in_ag(X489, T202))
ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)
F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
F129_IN_GA(T206, X511) → LESS71_IN_G(T206)
U25_GA(T206, X511, less71_out_g(T206)) → U26_GA(T206, X511, add102_in_ga(T206, X509))
U25_GA(T206, X511, less71_out_g(T206)) → ADD102_IN_GA(T206, X509)
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U28_GA(T206, X511, f129_in_ga(T210, X510))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → U30_GA(T206, X511, f129_in_aa(T213, X511))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T185, X419) → U23_AA(T185, X419, less59_in_a(T185))
F129_IN_AA(T185, X419) → LESS59_IN_A(T185)
LESS59_IN_A(s(T93)) → U9_A(T93, less10_in_a(T93))
LESS59_IN_A(s(T93)) → LESS10_IN_A(T93)
LESS10_IN_A(s(T19)) → U2_A(T19, less16_in_a(T19))
LESS10_IN_A(s(T19)) → LESS16_IN_A(T19)
LESS16_IN_A(s(T22)) → U1_A(T22, less16_in_a(T22))
LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)
LESS10_IN_A(s(T25)) → U3_A(T25, less10_in_a(T25))
LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)
LESS59_IN_A(s(T97)) → U10_A(T97, less59_in_a(T97))
LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)
U23_AA(T185, X419, less59_out_a(T185)) → U24_AA(T185, X419, add136_in_ag(X419, T185))
U23_AA(T185, X419, less59_out_a(T185)) → ADD136_IN_AG(X419, T185)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
F129_IN_AA(T206, X511) → LESS71_IN_A(T206)
LESS71_IN_A(s(T116)) → U21_A(T116, less77_in_a(T116))
LESS71_IN_A(s(T116)) → LESS77_IN_A(T116)
LESS77_IN_A(s(T122)) → U11_A(T122, less83_in_a(T122))
LESS77_IN_A(s(T122)) → LESS83_IN_A(T122)
LESS77_IN_A(T141) → U12_A(T141, less83_in_a(T141))
LESS77_IN_A(T141) → LESS83_IN_A(T141)
LESS71_IN_A(T145) → U22_A(T145, less77_in_a(T145))
LESS71_IN_A(T145) → LESS77_IN_A(T145)
U25_AA(T206, X511, less71_out_a(T206)) → U26_AA(T206, X511, add102_in_ga(T206, X509))
U25_AA(T206, X511, less71_out_a(T206)) → ADD102_IN_GA(T206, X509)
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U28_AA(T206, X511, f129_in_ga(T210, X510))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → U30_AA(T206, X511, f129_in_aa(T213, X511))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U39_GA(T107, T109, f129_in_ga(T148, T177))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → U40_GA(T107, T109, f1_in_aa(T177, T109))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(s(T12), T7) → U31_AA(T12, T7, less10_in_a(T12))
F1_IN_AA(s(T12), T7) → LESS10_IN_A(T12)
U31_AA(T12, T7, less10_out_a(T12)) → U32_AA(T12, T7, add11_in_ag(T7, T12))
U31_AA(T12, T7, less10_out_a(T12)) → ADD11_IN_AG(T7, T12)
F1_IN_AA(s(T86), T7) → U33_AA(T86, T7, less59_in_a(T86))
F1_IN_AA(s(T86), T7) → LESS59_IN_A(T86)
U33_AA(T86, T7, less59_out_a(T86)) → U34_AA(T86, T7, add11_in_ag(T7, T86))
U33_AA(T86, T7, less59_out_a(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
F1_IN_AA(T107, T109) → LESS71_IN_A(T107)
U35_AA(T107, T109, less71_out_a(T107)) → U36_AA(T107, T109, add102_in_ga(T107, X193))
U35_AA(T107, T109, less71_out_a(T107)) → ADD102_IN_GA(T107, X193)
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U38_AA(T107, T109, f129_in_ga(T148, X194))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → U40_AA(T107, T109, f1_in_aa(T177, T109))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)
LESS16_IN_G(x1)  =  LESS16_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)
U32_GA(x1, x2, x3)  =  U32_GA(x3)
ADD11_IN_AG(x1, x2)  =  ADD11_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x3)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x3)
U8_AG(x1, x2, x3)  =  U8_AG(x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)
U9_G(x1, x2)  =  U9_G(x2)
U10_G(x1, x2)  =  U10_G(x2)
U34_GA(x1, x2, x3)  =  U34_GA(x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
LESS71_IN_G(x1)  =  LESS71_IN_G(x1)
U21_G(x1, x2)  =  U21_G(x2)
LESS77_IN_G(x1)  =  LESS77_IN_G(x1)
U11_G(x1, x2)  =  U11_G(x2)
LESS83_IN_G(x1)  =  LESS83_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x2)
U22_G(x1, x2)  =  U22_G(x2)
U36_GA(x1, x2, x3)  =  U36_GA(x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)
U13_GA(x1, x2, x3)  =  U13_GA(x3)
U14_GA(x1, x2, x3)  =  U14_GA(x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x3)
U16_GA(x1, x2, x3)  =  U16_GA(x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)
U17_GA(x1, x2, x3)  =  U17_GA(x3)
U37_GA(x1, x2, x3)  =  U37_GA(x3)
U38_GA(x1, x2, x3)  =  U38_GA(x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)
U18_AG(x1, x2, x3)  =  U18_AG(x3)
U19_AG(x1, x2, x3)  =  U19_AG(x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x3)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U26_GA(x1, x2, x3)  =  U26_GA(x3)
U27_GA(x1, x2, x3)  =  U27_GA(x3)
U28_GA(x1, x2, x3)  =  U28_GA(x3)
U29_GA(x1, x2, x3)  =  U29_GA(x3)
U30_GA(x1, x2, x3)  =  U30_GA(x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U23_AA(x1, x2, x3)  =  U23_AA(x3)
LESS59_IN_A(x1)  =  LESS59_IN_A
U9_A(x1, x2)  =  U9_A(x2)
LESS10_IN_A(x1)  =  LESS10_IN_A
U2_A(x1, x2)  =  U2_A(x2)
LESS16_IN_A(x1)  =  LESS16_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U10_A(x1, x2)  =  U10_A(x2)
U24_AA(x1, x2, x3)  =  U24_AA(x1, x3)
U25_AA(x1, x2, x3)  =  U25_AA(x3)
LESS71_IN_A(x1)  =  LESS71_IN_A
U21_A(x1, x2)  =  U21_A(x2)
LESS77_IN_A(x1)  =  LESS77_IN_A
U11_A(x1, x2)  =  U11_A(x2)
LESS83_IN_A(x1)  =  LESS83_IN_A
U12_A(x1, x2)  =  U12_A(x2)
U22_A(x1, x2)  =  U22_A(x2)
U26_AA(x1, x2, x3)  =  U26_AA(x1, x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U28_AA(x1, x2, x3)  =  U28_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)
U30_AA(x1, x2, x3)  =  U30_AA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x3)
U40_GA(x1, x2, x3)  =  U40_GA(x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U31_AA(x1, x2, x3)  =  U31_AA(x3)
U32_AA(x1, x2, x3)  =  U32_AA(x1, x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3)  =  U34_AA(x1, x3)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U36_AA(x1, x2, x3)  =  U36_AA(x1, x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U38_AA(x1, x2, x3)  =  U38_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)
U40_AA(x1, x2, x3)  =  U40_AA(x1, x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 15 SCCs with 99 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS16_IN_A(x1)  =  LESS16_IN_A

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:

LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS16_IN_A(x1)  =  LESS16_IN_A

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:

LESS16_IN_A → LESS16_IN_A

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

(14) 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 = LESS16_IN_A evaluates to t =LESS16_IN_A

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 LESS16_IN_A to LESS16_IN_A.

(15) NO

(16) Obligation:

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

LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS10_IN_A(x1)  =  LESS10_IN_A

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:

LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS10_IN_A(x1)  =  LESS10_IN_A

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:

LESS10_IN_A → LESS10_IN_A

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

(21) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS10_IN_A evaluates to t =LESS10_IN_A

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 LESS10_IN_A to LESS10_IN_A.

(22) NO

(23) Obligation:

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

LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS59_IN_A(x1)  =  LESS59_IN_A

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:

LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS59_IN_A(x1)  =  LESS59_IN_A

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:

LESS59_IN_A → LESS59_IN_A

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 = LESS59_IN_A evaluates to t =LESS59_IN_A

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 LESS59_IN_A to LESS59_IN_A.

(29) NO

(30) Obligation:

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

ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)

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:

ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)

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:

ADD143_IN_AG(s(T202)) → ADD143_IN_AG(T202)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• ADD143_IN_AG(s(T202)) → ADD143_IN_AG(T202)
  The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

ADD136_IN_AG(s(T193)) → ADD136_IN_AG(T193)

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

(42) 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:

• ADD136_IN_AG(s(T193)) → ADD136_IN_AG(T193)
  The graph contains the following edges 1 > 1

(43) YES

(44) Obligation:

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

ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_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:

ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD117_IN_GA(x1, x2)  =  ADD117_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:

ADD117_IN_GA(s(T174)) → ADD117_IN_GA(T174)

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:

• ADD117_IN_GA(s(T174)) → ADD117_IN_GA(T174)
  The graph contains the following edges 1 > 1

(50) YES

(51) Obligation:

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

ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_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:

ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD110_IN_GA(x1, x2)  =  ADD110_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:

ADD110_IN_GA(s(T165)) → ADD110_IN_GA(T165)

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:

• ADD110_IN_GA(s(T165)) → ADD110_IN_GA(T165)
  The graph contains the following edges 1 > 1

(57) YES

(58) Obligation:

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

ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)

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

(59) UsableRulesProof (EQUIVALENT transformation)

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

(60) Obligation:

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

ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)

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

(61) PiDPToQDPProof (SOUND transformation)

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

(62) Obligation:

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

ADD102_IN_GA(s(T154)) → ADD102_IN_GA(T154)

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

(63) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• ADD102_IN_GA(s(T154)) → ADD102_IN_GA(T154)
  The graph contains the following edges 1 > 1

(64) YES

(65) Obligation:

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

ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

ADD41_IN_AG(s(T71)) → ADD41_IN_AG(T71)

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

(70) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• ADD41_IN_AG(s(T71)) → ADD41_IN_AG(T71)
  The graph contains the following edges 1 > 1

(71) YES

(72) Obligation:

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

ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

ADD33_IN_AG(s(T51)) → ADD33_IN_AG(T51)

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

(77) 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:

• ADD33_IN_AG(s(T51)) → ADD33_IN_AG(T51)
  The graph contains the following edges 1 > 1

(78) YES

(79) Obligation:

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

LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS16_IN_G(x1)  =  LESS16_IN_G(x1)

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

(80) UsableRulesProof (EQUIVALENT transformation)

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

(81) Obligation:

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

LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)

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

(82) PiDPToQDPProof (EQUIVALENT transformation)

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

(83) Obligation:

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

LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)

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

(84) 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:

• LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)
  The graph contains the following edges 1 > 1

(85) YES

(86) Obligation:

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

LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)

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

(87) UsableRulesProof (EQUIVALENT transformation)

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

(88) Obligation:

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

LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)

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

(89) PiDPToQDPProof (EQUIVALENT transformation)

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

(90) Obligation:

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

LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)

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

(91) 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:

• LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)
  The graph contains the following edges 1 > 1

(92) YES

(93) Obligation:

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

LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)

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

(94) UsableRulesProof (EQUIVALENT transformation)

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

(95) Obligation:

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

LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)

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

(96) PiDPToQDPProof (EQUIVALENT transformation)

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

(97) Obligation:

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

LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)

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

(98) 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:

• LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)
  The graph contains the following edges 1 > 1

(99) YES

(100) Obligation:

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

F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U27_GA(x1, x2, x3)  =  U27_GA(x3)
U29_GA(x1, x2, x3)  =  U29_GA(x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U25_AA(x1, x2, x3)  =  U25_AA(x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)

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

(101) UsableRulesProof (EQUIVALENT transformation)

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

(102) Obligation:

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

F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
less83_in_a(0) → less83_out_a(0)
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U27_GA(x1, x2, x3)  =  U27_GA(x3)
U29_GA(x1, x2, x3)  =  U29_GA(x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U25_AA(x1, x2, x3)  =  U25_AA(x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)

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

(103) PiDPToQDPProof (SOUND transformation)

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

(104) Obligation:

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

F129_IN_GA(T206) → U25_GA(T206, less71_in_g(T206))
U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_in_a)
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(105) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(T206) → U25_GA(T206, less71_in_g(T206)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(less77_in_g(x0)))
F129_IN_GA(x0) → U25_GA(x0, U22_g(less77_in_g(x0)))

(106) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_in_a)
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(less77_in_g(x0)))
F129_IN_GA(x0) → U25_GA(x0, U22_g(less77_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(107) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_AA → U25_AA(less71_in_a) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))

(108) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(less77_in_g(x0)))
F129_IN_GA(x0) → U25_GA(x0, U22_g(less77_in_g(x0)))
F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(109) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(less77_in_g(x0))) at position [1,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))

(110) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(less77_in_g(x0)))
F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(111) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(x0) → U25_GA(x0, U22_g(less77_in_g(x0))) at position [1,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))

(112) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(113) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(less77_in_a)) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a)))

(114) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(115) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U11_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0))))

(116) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a)))
F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(117) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0))))

(118) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(119) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0))))

(120) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(121) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(less77_out_a(0)))

(122) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(123) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0)))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(s(s(0))))

(124) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(0)))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(125) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(less77_out_a(0))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(s(0)))

(126) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(127) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(less77_in_a)) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a)))

(128) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(129) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U11_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0))))

(130) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a)))
F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(131) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0))))

(132) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(133) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0))))

(134) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(135) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(less77_out_a(0)))

(136) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0))))
F129_IN_AA → U25_AA(U22_a(less77_out_a(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(137) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0)))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(s(0)))

(138) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(less77_out_a(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(139) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(less77_out_a(0))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(0))

(140) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(141) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(U11_g(less83_out_g)))

(142) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(U11_g(less83_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(143) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(U11_g(less83_out_g))) at position [1,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(less77_out_g))

(144) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(less77_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(145) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(less77_out_g)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)

(146) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(147) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(U12_g(less83_in_g(x0)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(U12_g(less83_out_g)))

(148) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(U12_g(less83_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(149) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(U12_g(less83_out_g))) at position [1,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(less77_out_g))

(150) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(less77_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(151) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(less77_out_g)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)

(152) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(153) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(U11_g(less83_in_g(x0)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(U11_g(less83_out_g)))

(154) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(U11_g(less83_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(155) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(U11_g(less83_out_g))) at position [1,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(less77_out_g))

(156) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(less77_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(157) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(less77_out_g)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)

(158) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(159) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(x0) → U25_GA(x0, U22_g(U12_g(less83_in_g(x0)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U25_GA(0, U22_g(U12_g(less83_out_g)))

(160) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, U22_g(U12_g(less83_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(161) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(0) → U25_GA(0, U22_g(U12_g(less83_out_g))) at position [1,0] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U25_GA(0, U22_g(less77_out_g))

(162) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, U22_g(less77_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(163) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(0) → U25_GA(0, U22_g(less77_out_g)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U25_GA(0, less71_out_g)

(164) Obligation:

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

U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, less71_out_g)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(165) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_GA(T206, less71_out_g) → U27_GA(add102_in_ga(T206)) we obtained the following new rules [LPAR04]:

U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))

(166) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(167) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206)) we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))

(168) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(169) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(T206, add102_out_ga(T210)) → F129_IN_GA(T210) we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)

(170) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(171) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(T206, add102_out_ga(T210)) → U29_AA(T206, f129_in_ga(T210)) we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))

(172) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
U29_AA(T206, f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(173) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U29_AA(T206, f129_out_ga) → F129_IN_AA we obtained the following new rules [LPAR04]:

U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
U29_AA(s(0), f129_out_ga) → F129_IN_AA
U29_AA(0, f129_out_ga) → F129_IN_AA

(174) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
F129_IN_GA(0) → U25_GA(0, less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
U29_AA(s(0), f129_out_ga) → F129_IN_AA
U29_AA(0, f129_out_ga) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(175) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F129_IN_GA(0) → U25_GA(0, less71_out_g)

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

POL(0) = 1   
POL(F129_IN_AA) = 0   
POL(F129_IN_GA(x₁)) = x₁   
POL(U10_a(x₁)) = 0   
POL(U10_g(x₁)) = 0   
POL(U11_a(x₁)) = 0   
POL(U11_g(x₁)) = 0   
POL(U12_a(x₁)) = 0   
POL(U12_g(x₁)) = 1   
POL(U13_ga(x₁)) = 0   
POL(U14_ga(x₁)) = 0   
POL(U15_ga(x₁)) = 0   
POL(U16_ga(x₁)) = 0   
POL(U17_ga(x₁)) = 0   
POL(U18_ag(x₁)) = 0   
POL(U19_ag(x₁)) = 0   
POL(U1_a(x₁)) = 0   
POL(U1_g(x₁)) = 0   
POL(U20_ag(x₁)) = 0   
POL(U21_a(x₁)) = 0   
POL(U21_g(x₁)) = 0   
POL(U22_a(x₁)) = 0   
POL(U22_g(x₁)) = 0   
POL(U23_aa(x₁)) = 0   
POL(U23_ga(x₁, x₂)) = 0   
POL(U24_aa(x₁, x₂)) = 0   
POL(U24_ga(x₁)) = 0   
POL(U25_AA(x₁)) = 0   
POL(U25_GA(x₁, x₂)) = 0   
POL(U25_aa(x₁)) = 0   
POL(U25_ga(x₁, x₂)) = 0   
POL(U26_aa(x₁, x₂)) = 0   
POL(U26_ga(x₁)) = 0   
POL(U27_AA(x₁, x₂)) = x₂   
POL(U27_GA(x₁)) = x₁   
POL(U27_aa(x₁, x₂)) = 0   
POL(U27_ga(x₁)) = 0   
POL(U28_aa(x₁, x₂)) = 0   
POL(U28_ga(x₁)) = 0   
POL(U29_AA(x₁, x₂)) = 0   
POL(U29_GA(x₁)) = 0   
POL(U29_aa(x₁, x₂)) = 0   
POL(U29_ga(x₁)) = 0   
POL(U2_a(x₁)) = 0   
POL(U2_g(x₁)) = 0   
POL(U30_aa(x₁, x₂)) = 0   
POL(U30_ga(x₁)) = 0   
POL(U3_a(x₁)) = 0   
POL(U3_g(x₁)) = 0   
POL(U9_a(x₁)) = 0   
POL(U9_g(x₁)) = 0   
POL(add102_in_ga(x₁)) = 0   
POL(add102_out_ga(x₁)) = x₁   
POL(add110_in_ga(x₁)) = 0   
POL(add110_out_ga(x₁)) = 0   
POL(add117_in_ga(x₁)) = 0   
POL(add117_out_ga(x₁)) = 0   
POL(add136_in_ag(x₁)) = 0   
POL(add136_out_ag(x₁)) = 0   
POL(add143_in_ag(x₁)) = 1 + x₁   
POL(add143_out_ag(x₁)) = 0   
POL(f129_in_aa) = 0   
POL(f129_in_ga(x₁)) = 0   
POL(f129_out_aa(x₁)) = 0   
POL(f129_out_ga) = 0   
POL(less10_in_a) = 0   
POL(less10_in_g(x₁)) = 0   
POL(less10_out_a(x₁)) = 0   
POL(less10_out_g) = 0   
POL(less16_in_a) = 0   
POL(less16_in_g(x₁)) = 0   
POL(less16_out_a(x₁)) = 0   
POL(less16_out_g) = 0   
POL(less59_in_a) = 0   
POL(less59_in_g(x₁)) = 0   
POL(less59_out_a(x₁)) = 0   
POL(less59_out_g) = 0   
POL(less71_in_a) = 0   
POL(less71_in_g(x₁)) = 0   
POL(less71_out_a(x₁)) = 0   
POL(less71_out_g) = 0   
POL(less77_in_a) = 0   
POL(less77_in_g(x₁)) = 1   
POL(less77_out_a(x₁)) = 0   
POL(less77_out_g) = 0   
POL(less83_in_a) = 0   
POL(less83_in_g(x₁)) = 1   
POL(less83_out_a(x₁)) = 0   
POL(less83_out_g) = 1   
POL(s(x₁)) = 0   

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))

(176) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U25_GA(0, less71_out_g) → U27_GA(add102_in_ga(0))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
U29_AA(s(0), f129_out_ga) → F129_IN_AA
U29_AA(0, f129_out_ga) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(177) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(178) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g)

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

POL(F129_IN_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(0) =

/ 1 \ \ 0 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(less71_out_g) =

/ 0 \ \ 1 /

POL(U27_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(add102_in_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add102_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(U29_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(f129_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(f129_out_ga) =

/ 0 \ \ 1 /

POL(F129_IN_AA) =

1

POL(U25_AA(x1)) =

1

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U13_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U14_ga(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(add110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(less71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U27_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U28_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U21_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(U22_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U26_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U29_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U30_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(f129_in_aa) =

/ 0 \ \ 1 /

POL(U23_aa(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 0 /

POL(f129_out_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 0 \ \ 1 /

POL(U21_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 0 \ \ 0 /

POL(U22_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U26_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U30_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U11_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(less83_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U12_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(less77_out_g) =

/ 1 \ \ 0 /

POL(U15_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U16_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add110_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U9_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_in_g(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U10_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_out_g) =

/ 1 \ \ 0 /

POL(U24_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add136_in_ag(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U11_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less83_in_a) =

/ 0 \ \ 1 /

POL(U12_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less77_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U18_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U19_ag(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(add143_in_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(add136_out_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less83_out_g) =

/ 0 \ \ 1 /

POL(add117_out_ga(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(U17_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U2_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less16_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U3_g(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less10_out_g) =

/ 0 \ \ 0 /

POL(less83_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U9_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_in_a) =

/ 0 \ \ 0 /

POL(U10_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less59_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U24_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(add143_out_ag(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U20_ag(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(less16_out_g) =

/ 1 \ \ 1 /

POL(U1_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U2_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less16_in_a) =

/ 0 \ \ 0 /

POL(U3_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(less10_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(less16_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U1_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U28_ga(f129_out_ga) → f129_out_ga
less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U30_ga(f129_out_aa(T213)) → f129_out_ga
f129_in_aa → U25_aa(less71_in_a)
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U26_ga(add102_out_ga(X509)) → f129_out_ga
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U24_ga(add136_out_ag(X419)) → f129_out_ga
less83_in_g(0) → less83_out_g
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U10_g(less59_out_g) → less59_out_g
U9_g(less10_out_g) → less59_out_g

(180) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U25_GA(s(0), less71_out_g) → U27_GA(add102_in_ga(s(0)))
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(181) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(182) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(183) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_GA(s(s(0)), less71_out_g) → U27_GA(add102_in_ga(s(s(0)))) at position [0] we obtained the following new rules [LPAR04]:

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))

(184) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(185) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(T210)) → U29_GA(f129_in_ga(T210)) at position [0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))

(186) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(187) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))

(188) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(189) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(x1)) → U29_AA(s(s(0)), f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))

(190) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(191) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0))) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))

(192) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(193) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(x1)) → U29_AA(s(0), f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))

(194) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(195) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0)) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(add110_in_ga(0)))

(196) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(add110_in_ga(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(197) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(add110_in_ga(0))) at position [1,0] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(U16_ga(add117_in_ga(0))))

(198) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(U16_ga(add117_in_ga(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(199) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(U16_ga(add117_in_ga(0)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(U16_ga(add117_out_ga(0))))

(200) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(U16_ga(add117_out_ga(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(201) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(U16_ga(add117_out_ga(0)))) at position [1,0] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(add110_out_ga(s(0))))

(202) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(add110_out_ga(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(203) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(add110_out_ga(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))

(204) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(205) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(x1)) → U29_AA(0, f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))

(206) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(207) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, less71_in_g(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0))))

(208) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(209) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))

(210) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(211) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))

(212) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(213) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))

(214) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(215) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(less77_in_g(x0)))) at position [0,1,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

(216) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(217) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(less77_in_g(x0)))) at position [0,1,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

(218) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(219) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(less77_in_g(x0)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

(220) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(221) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(less77_in_g(x0)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

(222) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(223) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(less77_in_g(x0)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

(224) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(225) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(less77_in_g(x0)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

(226) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(227) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(less77_in_g(x0)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

(228) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0))))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(229) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(less77_in_g(x0)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

(230) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(231) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(s(s(x0)))) → U29_GA(U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

(232) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(233) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), U21_g(U11_g(less83_out_g)))) at position [0,1,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), U21_g(less77_out_g)))

(234) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(235) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), U21_g(less77_out_g))) at position [0,1] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))

(236) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(237) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0))))) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U21_g(U12_g(less83_out_g))))

(238) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U21_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(239) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U21_g(U12_g(less83_out_g)))) at position [0,1,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U21_g(less77_out_g)))

(240) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(241) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U21_g(less77_out_g))) at position [0,1] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

(242) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(243) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(s(x0))) → U29_GA(U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0))))) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U22_g(U11_g(less83_out_g))))

(244) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U22_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(245) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U22_g(U11_g(less83_out_g)))) at position [0,1,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U22_g(less77_out_g)))

(246) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U22_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(247) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), U22_g(less77_out_g))) at position [0,1] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

(248) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(249) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(add102_out_ga(x0)) → U29_GA(U25_ga(x0, U22_g(U12_g(less83_in_g(x0))))) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(0)) → U29_GA(U25_ga(0, U22_g(U12_g(less83_out_g))))

(250) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga) → F129_IN_AA
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_GA(add102_out_ga(0)) → U29_GA(U25_ga(0, U22_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(251) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(252) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(253) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

(254) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(255) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(U11_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(less77_out_g)))

(256) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(257) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))

(258) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(259) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(0))) → U29_AA(0, U25_ga(s(0), U21_g(U12_g(less83_out_g))))

(260) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(0, add102_out_ga(s(0))) → U29_AA(0, U25_ga(s(0), U21_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(261) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(262) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(263) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(0))) → U29_AA(0, U25_ga(s(0), U22_g(U11_g(less83_out_g))))

(264) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(0, add102_out_ga(s(0))) → U29_AA(0, U25_ga(s(0), U22_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(265) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(266) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(267) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U25_ga(x0, U22_g(U12_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0)) → U29_AA(0, U25_ga(0, U22_g(U12_g(less83_out_g))))

(268) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0)) → U29_AA(0, U25_ga(0, U22_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(269) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(270) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(271) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

(272) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(273) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(U11_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(less77_out_g)))

(274) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(275) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))

(276) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(277) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(U12_g(less83_out_g))))

(278) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(279) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(U12_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(less77_out_g)))

(280) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(281) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))

(282) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(283) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(U11_g(less83_out_g))))

(284) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(285) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(U11_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(less77_out_g)))

(286) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(287) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))

(288) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(289) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U25_ga(x0, U22_g(U12_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(0)) → U29_AA(s(0), U25_ga(0, U22_g(U12_g(less83_out_g))))

(290) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U27_AA(s(0), add102_out_ga(0)) → U29_AA(s(0), U25_ga(0, U22_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(291) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(292) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(293) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(U11_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

(294) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(295) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(U11_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(less77_out_g)))

(296) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(297) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))

(298) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(299) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(U12_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(U12_g(less83_out_g))))

(300) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(301) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(U12_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(less77_out_g)))

(302) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(303) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))

(304) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(305) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(U11_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(U11_g(less83_out_g))))

(306) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(U11_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(307) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(U11_g(less83_out_g)))) at position [1,1,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(less77_out_g)))

(308) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(less77_out_g)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(309) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(less77_out_g))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))

(310) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(311) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(U12_g(less83_in_g(x0))))) at position [1,1,0,0] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(0)) → U29_AA(s(s(0)), U25_ga(0, U22_g(U12_g(less83_out_g))))

(312) Obligation:

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

U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(0)) → U29_AA(s(s(0)), U25_ga(0, U22_g(U12_g(less83_out_g))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(313) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(314) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(315) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(0, add102_out_ga(x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

(316) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(317) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(0, add102_out_ga(x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0))) we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))

(318) Obligation:

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

U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(319) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_GA(add102_out_ga(T210)) → F129_IN_GA(T210) we obtained the following new rules [LPAR04]:

U27_GA(add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

(320) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(321) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_AA(s(s(0)), add102_out_ga(x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

(322) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(323) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_AA(s(0), add102_out_ga(x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

(324) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(325) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U27_GA(add102_out_ga(s(0))) → U29_GA(U25_ga(s(0), less71_out_g))

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

POL(F129_IN_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(0) =

/ 1 \ \ 0 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(less71_out_g) =

/ 0 \ \ 0 /

POL(U27_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(U13_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(add102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(add102_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U29_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(f129_out_ga) =

/ 1 \ \ 1 /

POL(F129_IN_AA) =

0

POL(U25_AA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(add110_in_ga(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(U9_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U10_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_out_g) =

/ 1 \ \ 0 /

POL(U24_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(add136_in_ag(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL(U26_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U27_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U15_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U16_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add117_in_ga(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(add110_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U28_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(f129_in_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(less71_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U21_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U22_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U29_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U30_ga(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(f129_in_aa) =

/ 1 \ \ 1 /

POL(U23_aa(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 0 /

POL(f129_out_aa(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 1 \ \ 0 /

POL(U21_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 0 \ \ 0 /

POL(U22_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 1 0 /

·

x2

POL(U30_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U11_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(less83_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U12_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_out_g) =

/ 0 \ \ 0 /

POL(U11_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less83_in_a) =

/ 0 \ \ 0 /

POL(U12_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U18_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U19_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add143_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add136_out_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less83_out_g) =

/ 1 \ \ 1 /

POL(add117_out_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 1 /

·

x1

POL(U17_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U2_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(less16_in_g(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U3_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(less10_out_g) =

/ 0 \ \ 0 /

POL(less83_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(less10_in_a) =

/ 1 \ \ 1 /

POL(U10_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(U24_aa(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_out_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U20_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less16_out_g) =

/ 0 \ \ 0 /

POL(U1_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U2_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less16_in_a) =

/ 1 \ \ 1 /

POL(U3_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less16_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U1_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))

(326) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(327) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U27_AA(s(0), add102_out_ga(s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g))
U27_AA(s(s(0)), add102_out_ga(s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g))

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

POL(F129_IN_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(0) =

/ 0 \ \ 1 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(less71_out_g) =

/ 1 \ \ 1 /

POL(U27_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(U13_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(add102_in_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add102_out_ga(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(U29_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(f129_out_ga) =

/ 1 \ \ 1 /

POL(F129_IN_AA) =

0

POL(U25_AA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(add110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U10_g(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less59_out_g) =

/ 1 \ \ 1 /

POL(U24_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL(add136_in_ag(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U15_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U16_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(add110_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U28_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(f129_in_ga(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U21_g(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less77_in_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U29_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U30_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(f129_in_aa) =

/ 1 \ \ 0 /

POL(U23_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 0 /

POL(f129_out_aa(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 1 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 1 \ \ 0 /

POL(U21_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 0 \ \ 0 /

POL(U22_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U30_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U11_g(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less83_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U12_g(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less77_out_g) =

/ 0 \ \ 0 /

POL(U11_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less83_in_a) =

/ 0 \ \ 0 /

POL(U12_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U18_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U19_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add143_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(add136_out_ag(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less83_out_g) =

/ 1 \ \ 1 /

POL(add117_out_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U17_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U2_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(less16_in_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(U3_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_out_g) =

/ 1 \ \ 1 /

POL(less83_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U9_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less10_in_a) =

/ 0 \ \ 0 /

POL(U10_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(less59_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U24_aa(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(add143_out_ag(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U20_ag(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less16_out_g) =

/ 0 \ \ 0 /

POL(U1_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U2_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(less16_in_a) =

/ 1 \ \ 1 /

POL(U3_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less10_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less16_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U1_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))

(328) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g)
U25_GA(s(s(0)), less71_out_g) → U27_GA(U13_ga(add102_in_ga(s(0))))
U27_GA(add102_out_ga(x0)) → U29_GA(U23_ga(x0, less59_in_g(x0)))
U29_GA(f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(s(s(0))))
U29_AA(0, f129_out_ga) → F129_IN_AA
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g))
U27_AA(s(0), add102_out_ga(s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(add110_in_ga(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → U29_GA(U25_ga(s(s(0)), less71_out_g))
U25_GA(s(s(0)), less71_out_g) → U27_GA(U14_ga(add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(less77_in_g(T116))
less71_in_g(T145) → U22_g(less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(less77_out_g) → less71_out_g
U22_g(less77_out_g) → less71_out_g
U13_ga(add102_out_ga(X302)) → add102_out_ga(s(X302))
U14_ga(add110_out_ga(X325)) → add102_out_ga(s(X325))
U23_ga(T185, less59_out_g) → f129_out_ga
U23_ga(T185, less59_out_g) → U24_ga(add136_in_ag(T185))
U25_ga(T206, less71_out_g) → f129_out_ga
U25_ga(T206, less71_out_g) → U26_ga(add102_in_ga(T206))
U25_ga(T206, less71_out_g) → U27_ga(add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(less83_in_g(T122))
less77_in_g(T141) → U12_g(less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(less59_in_g(T97))
U24_ga(add136_out_ag(X419)) → f129_out_ga
U26_ga(add102_out_ga(X509)) → f129_out_ga
U27_ga(add102_out_ga(T210)) → U28_ga(f129_in_ga(T210))
U27_ga(add102_out_ga(T210)) → U29_ga(f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(less83_out_g) → less77_out_g
U12_g(less83_out_g) → less77_out_g
U15_ga(add110_out_ga(X349)) → add110_out_ga(s(X349))
U16_ga(add117_out_ga(X368)) → add110_out_ga(s(X368))
U9_g(less10_out_g) → less59_out_g
U10_g(less59_out_g) → less59_out_g
add136_in_ag(s(T193)) → U18_ag(add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(add143_in_ag(T197))
U28_ga(f129_out_ga) → f129_out_ga
U29_ga(f129_out_ga) → U30_ga(f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g
add117_in_ga(0) → add117_out_ga(0)
add117_in_ga(s(T174)) → U17_ga(add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(less10_in_g(T25))
U18_ag(add136_out_ag(X447)) → add136_out_ag(s(X447))
U19_ag(add143_out_ag(X466)) → add136_out_ag(X466)
U30_ga(f129_out_aa(T213)) → f129_out_ga
less83_in_a → less83_out_a(0)
U17_ga(add117_out_ga(X388)) → add117_out_ga(s(X388))
U2_g(less16_out_g) → less10_out_g
U3_g(less10_out_g) → less10_out_g
add143_in_ag(0) → add143_out_ag(0)
add143_in_ag(s(T202)) → U20_ag(add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g
less16_in_g(s(T22)) → U1_g(less16_in_g(T22))
U20_ag(add143_out_ag(X489)) → add143_out_ag(s(X489))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(less16_out_g) → less16_out_g
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0)
U22_g(x0)
U13_ga(x0)
U14_ga(x0)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0)
U26_ga(x0)
U27_ga(x0)
less77_in_a
U11_g(x0)
U12_g(x0)
U15_ga(x0)
U16_ga(x0)
U9_g(x0)
U10_g(x0)
add136_in_ag(x0)
U28_ga(x0)
U29_ga(x0)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0)
U19_ag(x0)
U30_ga(x0)
less83_in_a
U17_ga(x0)
U2_g(x0)
U3_g(x0)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0)
U23_aa(x0)
U25_aa(x0)
U1_g(x0)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(329) Obligation:

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

U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
f1_out_ga(x1, x2)  =  f1_out_ga
U32_ga(x1, x2, x3)  =  U32_ga(x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1)
add11_out_ag(x1, x2)  =  add11_out_ag(x1)
U8_ag(x1, x2, x3)  =  U8_ag(x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
U34_ga(x1, x2, x3)  =  U34_ga(x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
U36_ga(x1, x2, x3)  =  U36_ga(x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
U37_ga(x1, x2, x3)  =  U37_ga(x3)
U38_ga(x1, x2, x3)  =  U38_ga(x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x3)
U40_ga(x1, x2, x3)  =  U40_ga(x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)

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

(330) UsableRulesProof (EQUIVALENT transformation)

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

(331) Obligation:

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

U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))

The TRS R consists of the following rules:

f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
less83_in_a(0) → less83_out_a(0)
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
less83_in_g(0) → less83_out_g(0)
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g
U1_g(x1, x2)  =  U1_g(x2)
less10_out_g(x1)  =  less10_out_g
U3_g(x1, x2)  =  U3_g(x2)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x2)
less59_out_g(x1)  =  less59_out_g
U10_g(x1, x2)  =  U10_g(x2)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g
less77_out_g(x1)  =  less77_out_g
U12_g(x1, x2)  =  U12_g(x2)
less71_out_g(x1)  =  less71_out_g
U22_g(x1, x2)  =  U22_g(x2)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x3)
U14_ga(x1, x2, x3)  =  U14_ga(x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x3)
U16_ga(x1, x2, x3)  =  U16_ga(x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x2)
U17_ga(x1, x2, x3)  =  U17_ga(x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x2)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga
U24_ga(x1, x2, x3)  =  U24_ga(x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x3)
U19_ag(x1, x2, x3)  =  U19_ag(x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x3)
U27_ga(x1, x2, x3)  =  U27_ga(x3)
U28_ga(x1, x2, x3)  =  U28_ga(x3)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U30_ga(x1, x2, x3)  =  U30_ga(x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)

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

(332) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
f1_in: (b,f) (f,f)
less10_in: (b) (f)
less16_in: (b) (f)
add11_in: (f,b)
add33_in: (f,b)
add41_in: (f,b)
less59_in: (b) (f)
less71_in: (b) (f)
less77_in: (b) (f)
add102_in: (b,f)
add110_in: (b,f)
add117_in: (b,f)
f129_in: (b,f) (f,f)
add136_in: (f,b)
add143_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(333) Obligation:

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

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)

(334) 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:

F1_IN_GA(s(T12), T7) → U31_GA(T12, T7, less10_in_g(T12))
F1_IN_GA(s(T12), T7) → LESS10_IN_G(T12)
LESS10_IN_G(s(T19)) → U2_G(T19, less16_in_g(T19))
LESS10_IN_G(s(T19)) → LESS16_IN_G(T19)
LESS16_IN_G(s(T22)) → U1_G(T22, less16_in_g(T22))
LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)
LESS10_IN_G(s(T25)) → U3_G(T25, less10_in_g(T25))
LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)
U31_GA(T12, T7, less10_out_g(T12)) → U32_GA(T12, T7, add11_in_ag(T7, T12))
U31_GA(T12, T7, less10_out_g(T12)) → ADD11_IN_AG(T7, T12)
ADD11_IN_AG(s(T39), T38) → U7_AG(T39, T38, add33_in_ag(T39, T38))
ADD11_IN_AG(s(T39), T38) → ADD33_IN_AG(T39, T38)
ADD33_IN_AG(s(T52), s(T51)) → U4_AG(T52, T51, add33_in_ag(T52, T51))
ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)
ADD33_IN_AG(T61, s(T60)) → U5_AG(T61, T60, add41_in_ag(T61, T60))
ADD33_IN_AG(T61, s(T60)) → ADD41_IN_AG(T61, T60)
ADD41_IN_AG(s(T72), s(T71)) → U6_AG(T72, T71, add41_in_ag(T72, T71))
ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)
ADD11_IN_AG(T81, T80) → U8_AG(T81, T80, add41_in_ag(T81, T80))
ADD11_IN_AG(T81, T80) → ADD41_IN_AG(T81, T80)
F1_IN_GA(s(T86), T7) → U33_GA(T86, T7, less59_in_g(T86))
F1_IN_GA(s(T86), T7) → LESS59_IN_G(T86)
LESS59_IN_G(s(T93)) → U9_G(T93, less10_in_g(T93))
LESS59_IN_G(s(T93)) → LESS10_IN_G(T93)
LESS59_IN_G(s(T97)) → U10_G(T97, less59_in_g(T97))
LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)
U33_GA(T86, T7, less59_out_g(T86)) → U34_GA(T86, T7, add11_in_ag(T7, T86))
U33_GA(T86, T7, less59_out_g(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_GA(T107, T109) → U35_GA(T107, T109, less71_in_g(T107))
F1_IN_GA(T107, T109) → LESS71_IN_G(T107)
LESS71_IN_G(s(T116)) → U21_G(T116, less77_in_g(T116))
LESS71_IN_G(s(T116)) → LESS77_IN_G(T116)
LESS77_IN_G(s(T122)) → U11_G(T122, less83_in_g(T122))
LESS77_IN_G(s(T122)) → LESS83_IN_G(T122)
LESS77_IN_G(T141) → U12_G(T141, less83_in_g(T141))
LESS77_IN_G(T141) → LESS83_IN_G(T141)
LESS71_IN_G(T145) → U22_G(T145, less77_in_g(T145))
LESS71_IN_G(T145) → LESS77_IN_G(T145)
U35_GA(T107, T109, less71_out_g(T107)) → U36_GA(T107, T109, add102_in_ga(T107, X193))
U35_GA(T107, T109, less71_out_g(T107)) → ADD102_IN_GA(T107, X193)
ADD102_IN_GA(s(T154), s(X302)) → U13_GA(T154, X302, add102_in_ga(T154, X302))
ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)
ADD102_IN_GA(T159, s(X325)) → U14_GA(T159, X325, add110_in_ga(T159, X325))
ADD102_IN_GA(T159, s(X325)) → ADD110_IN_GA(T159, X325)
ADD110_IN_GA(s(T165), s(X349)) → U15_GA(T165, X349, add110_in_ga(T165, X349))
ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)
ADD110_IN_GA(T169, s(X368)) → U16_GA(T169, X368, add117_in_ga(T169, X368))
ADD110_IN_GA(T169, s(X368)) → ADD117_IN_GA(T169, X368)
ADD117_IN_GA(s(T174), s(X388)) → U17_GA(T174, X388, add117_in_ga(T174, X388))
ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)
U35_GA(T107, T109, less71_out_g(T107)) → U37_GA(T107, T109, add102_in_ga(T107, T148))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U38_GA(T107, T109, f129_in_ga(T148, X194))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
F129_IN_GA(T185, X419) → U23_GA(T185, X419, less59_in_g(T185))
F129_IN_GA(T185, X419) → LESS59_IN_G(T185)
U23_GA(T185, X419, less59_out_g(T185)) → U24_GA(T185, X419, add136_in_ag(X419, T185))
U23_GA(T185, X419, less59_out_g(T185)) → ADD136_IN_AG(X419, T185)
ADD136_IN_AG(s(X447), s(T193)) → U18_AG(X447, T193, add136_in_ag(X447, T193))
ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)
ADD136_IN_AG(X466, s(T197)) → U19_AG(X466, T197, add143_in_ag(X466, T197))
ADD136_IN_AG(X466, s(T197)) → ADD143_IN_AG(X466, T197)
ADD143_IN_AG(s(X489), s(T202)) → U20_AG(X489, T202, add143_in_ag(X489, T202))
ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)
F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
F129_IN_GA(T206, X511) → LESS71_IN_G(T206)
U25_GA(T206, X511, less71_out_g(T206)) → U26_GA(T206, X511, add102_in_ga(T206, X509))
U25_GA(T206, X511, less71_out_g(T206)) → ADD102_IN_GA(T206, X509)
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U28_GA(T206, X511, f129_in_ga(T210, X510))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → U30_GA(T206, X511, f129_in_aa(T213, X511))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T185, X419) → U23_AA(T185, X419, less59_in_a(T185))
F129_IN_AA(T185, X419) → LESS59_IN_A(T185)
LESS59_IN_A(s(T93)) → U9_A(T93, less10_in_a(T93))
LESS59_IN_A(s(T93)) → LESS10_IN_A(T93)
LESS10_IN_A(s(T19)) → U2_A(T19, less16_in_a(T19))
LESS10_IN_A(s(T19)) → LESS16_IN_A(T19)
LESS16_IN_A(s(T22)) → U1_A(T22, less16_in_a(T22))
LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)
LESS10_IN_A(s(T25)) → U3_A(T25, less10_in_a(T25))
LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)
LESS59_IN_A(s(T97)) → U10_A(T97, less59_in_a(T97))
LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)
U23_AA(T185, X419, less59_out_a(T185)) → U24_AA(T185, X419, add136_in_ag(X419, T185))
U23_AA(T185, X419, less59_out_a(T185)) → ADD136_IN_AG(X419, T185)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
F129_IN_AA(T206, X511) → LESS71_IN_A(T206)
LESS71_IN_A(s(T116)) → U21_A(T116, less77_in_a(T116))
LESS71_IN_A(s(T116)) → LESS77_IN_A(T116)
LESS77_IN_A(s(T122)) → U11_A(T122, less83_in_a(T122))
LESS77_IN_A(s(T122)) → LESS83_IN_A(T122)
LESS77_IN_A(T141) → U12_A(T141, less83_in_a(T141))
LESS77_IN_A(T141) → LESS83_IN_A(T141)
LESS71_IN_A(T145) → U22_A(T145, less77_in_a(T145))
LESS71_IN_A(T145) → LESS77_IN_A(T145)
U25_AA(T206, X511, less71_out_a(T206)) → U26_AA(T206, X511, add102_in_ga(T206, X509))
U25_AA(T206, X511, less71_out_a(T206)) → ADD102_IN_GA(T206, X509)
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U28_AA(T206, X511, f129_in_ga(T210, X510))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → U30_AA(T206, X511, f129_in_aa(T213, X511))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U39_GA(T107, T109, f129_in_ga(T148, T177))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → U40_GA(T107, T109, f1_in_aa(T177, T109))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(s(T12), T7) → U31_AA(T12, T7, less10_in_a(T12))
F1_IN_AA(s(T12), T7) → LESS10_IN_A(T12)
U31_AA(T12, T7, less10_out_a(T12)) → U32_AA(T12, T7, add11_in_ag(T7, T12))
U31_AA(T12, T7, less10_out_a(T12)) → ADD11_IN_AG(T7, T12)
F1_IN_AA(s(T86), T7) → U33_AA(T86, T7, less59_in_a(T86))
F1_IN_AA(s(T86), T7) → LESS59_IN_A(T86)
U33_AA(T86, T7, less59_out_a(T86)) → U34_AA(T86, T7, add11_in_ag(T7, T86))
U33_AA(T86, T7, less59_out_a(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
F1_IN_AA(T107, T109) → LESS71_IN_A(T107)
U35_AA(T107, T109, less71_out_a(T107)) → U36_AA(T107, T109, add102_in_ga(T107, X193))
U35_AA(T107, T109, less71_out_a(T107)) → ADD102_IN_GA(T107, X193)
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U38_AA(T107, T109, f129_in_ga(T148, X194))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → U40_AA(T107, T109, f1_in_aa(T177, T109))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
LESS16_IN_G(x1)  =  LESS16_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_G(x1, x2)  =  U3_G(x1, x2)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
ADD11_IN_AG(x1, x2)  =  ADD11_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x2, x3)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
LESS71_IN_G(x1)  =  LESS71_IN_G(x1)
U21_G(x1, x2)  =  U21_G(x1, x2)
LESS77_IN_G(x1)  =  LESS77_IN_G(x1)
U11_G(x1, x2)  =  U11_G(x1, x2)
LESS83_IN_G(x1)  =  LESS83_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U22_G(x1, x2)  =  U22_G(x1, x2)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U37_GA(x1, x2, x3)  =  U37_GA(x1, x3)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)
U18_AG(x1, x2, x3)  =  U18_AG(x2, x3)
U19_AG(x1, x2, x3)  =  U19_AG(x2, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x2, x3)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U26_GA(x1, x2, x3)  =  U26_GA(x1, x3)
U27_GA(x1, x2, x3)  =  U27_GA(x1, x3)
U28_GA(x1, x2, x3)  =  U28_GA(x1, x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x1, x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U23_AA(x1, x2, x3)  =  U23_AA(x3)
LESS59_IN_A(x1)  =  LESS59_IN_A
U9_A(x1, x2)  =  U9_A(x2)
LESS10_IN_A(x1)  =  LESS10_IN_A
U2_A(x1, x2)  =  U2_A(x2)
LESS16_IN_A(x1)  =  LESS16_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U10_A(x1, x2)  =  U10_A(x2)
U24_AA(x1, x2, x3)  =  U24_AA(x1, x3)
U25_AA(x1, x2, x3)  =  U25_AA(x3)
LESS71_IN_A(x1)  =  LESS71_IN_A
U21_A(x1, x2)  =  U21_A(x2)
LESS77_IN_A(x1)  =  LESS77_IN_A
U11_A(x1, x2)  =  U11_A(x2)
LESS83_IN_A(x1)  =  LESS83_IN_A
U12_A(x1, x2)  =  U12_A(x2)
U22_A(x1, x2)  =  U22_A(x2)
U26_AA(x1, x2, x3)  =  U26_AA(x1, x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U28_AA(x1, x2, x3)  =  U28_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)
U30_AA(x1, x2, x3)  =  U30_AA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x1, x3)
U40_GA(x1, x2, x3)  =  U40_GA(x1, x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U31_AA(x1, x2, x3)  =  U31_AA(x3)
U32_AA(x1, x2, x3)  =  U32_AA(x1, x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3)  =  U34_AA(x1, x3)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U36_AA(x1, x2, x3)  =  U36_AA(x1, x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U38_AA(x1, x2, x3)  =  U38_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)
U40_AA(x1, x2, x3)  =  U40_AA(x1, x3)

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

(335) Obligation:

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

F1_IN_GA(s(T12), T7) → U31_GA(T12, T7, less10_in_g(T12))
F1_IN_GA(s(T12), T7) → LESS10_IN_G(T12)
LESS10_IN_G(s(T19)) → U2_G(T19, less16_in_g(T19))
LESS10_IN_G(s(T19)) → LESS16_IN_G(T19)
LESS16_IN_G(s(T22)) → U1_G(T22, less16_in_g(T22))
LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)
LESS10_IN_G(s(T25)) → U3_G(T25, less10_in_g(T25))
LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)
U31_GA(T12, T7, less10_out_g(T12)) → U32_GA(T12, T7, add11_in_ag(T7, T12))
U31_GA(T12, T7, less10_out_g(T12)) → ADD11_IN_AG(T7, T12)
ADD11_IN_AG(s(T39), T38) → U7_AG(T39, T38, add33_in_ag(T39, T38))
ADD11_IN_AG(s(T39), T38) → ADD33_IN_AG(T39, T38)
ADD33_IN_AG(s(T52), s(T51)) → U4_AG(T52, T51, add33_in_ag(T52, T51))
ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)
ADD33_IN_AG(T61, s(T60)) → U5_AG(T61, T60, add41_in_ag(T61, T60))
ADD33_IN_AG(T61, s(T60)) → ADD41_IN_AG(T61, T60)
ADD41_IN_AG(s(T72), s(T71)) → U6_AG(T72, T71, add41_in_ag(T72, T71))
ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)
ADD11_IN_AG(T81, T80) → U8_AG(T81, T80, add41_in_ag(T81, T80))
ADD11_IN_AG(T81, T80) → ADD41_IN_AG(T81, T80)
F1_IN_GA(s(T86), T7) → U33_GA(T86, T7, less59_in_g(T86))
F1_IN_GA(s(T86), T7) → LESS59_IN_G(T86)
LESS59_IN_G(s(T93)) → U9_G(T93, less10_in_g(T93))
LESS59_IN_G(s(T93)) → LESS10_IN_G(T93)
LESS59_IN_G(s(T97)) → U10_G(T97, less59_in_g(T97))
LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)
U33_GA(T86, T7, less59_out_g(T86)) → U34_GA(T86, T7, add11_in_ag(T7, T86))
U33_GA(T86, T7, less59_out_g(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_GA(T107, T109) → U35_GA(T107, T109, less71_in_g(T107))
F1_IN_GA(T107, T109) → LESS71_IN_G(T107)
LESS71_IN_G(s(T116)) → U21_G(T116, less77_in_g(T116))
LESS71_IN_G(s(T116)) → LESS77_IN_G(T116)
LESS77_IN_G(s(T122)) → U11_G(T122, less83_in_g(T122))
LESS77_IN_G(s(T122)) → LESS83_IN_G(T122)
LESS77_IN_G(T141) → U12_G(T141, less83_in_g(T141))
LESS77_IN_G(T141) → LESS83_IN_G(T141)
LESS71_IN_G(T145) → U22_G(T145, less77_in_g(T145))
LESS71_IN_G(T145) → LESS77_IN_G(T145)
U35_GA(T107, T109, less71_out_g(T107)) → U36_GA(T107, T109, add102_in_ga(T107, X193))
U35_GA(T107, T109, less71_out_g(T107)) → ADD102_IN_GA(T107, X193)
ADD102_IN_GA(s(T154), s(X302)) → U13_GA(T154, X302, add102_in_ga(T154, X302))
ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)
ADD102_IN_GA(T159, s(X325)) → U14_GA(T159, X325, add110_in_ga(T159, X325))
ADD102_IN_GA(T159, s(X325)) → ADD110_IN_GA(T159, X325)
ADD110_IN_GA(s(T165), s(X349)) → U15_GA(T165, X349, add110_in_ga(T165, X349))
ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)
ADD110_IN_GA(T169, s(X368)) → U16_GA(T169, X368, add117_in_ga(T169, X368))
ADD110_IN_GA(T169, s(X368)) → ADD117_IN_GA(T169, X368)
ADD117_IN_GA(s(T174), s(X388)) → U17_GA(T174, X388, add117_in_ga(T174, X388))
ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)
U35_GA(T107, T109, less71_out_g(T107)) → U37_GA(T107, T109, add102_in_ga(T107, T148))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U38_GA(T107, T109, f129_in_ga(T148, X194))
U37_GA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
F129_IN_GA(T185, X419) → U23_GA(T185, X419, less59_in_g(T185))
F129_IN_GA(T185, X419) → LESS59_IN_G(T185)
U23_GA(T185, X419, less59_out_g(T185)) → U24_GA(T185, X419, add136_in_ag(X419, T185))
U23_GA(T185, X419, less59_out_g(T185)) → ADD136_IN_AG(X419, T185)
ADD136_IN_AG(s(X447), s(T193)) → U18_AG(X447, T193, add136_in_ag(X447, T193))
ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)
ADD136_IN_AG(X466, s(T197)) → U19_AG(X466, T197, add143_in_ag(X466, T197))
ADD136_IN_AG(X466, s(T197)) → ADD143_IN_AG(X466, T197)
ADD143_IN_AG(s(X489), s(T202)) → U20_AG(X489, T202, add143_in_ag(X489, T202))
ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)
F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
F129_IN_GA(T206, X511) → LESS71_IN_G(T206)
U25_GA(T206, X511, less71_out_g(T206)) → U26_GA(T206, X511, add102_in_ga(T206, X509))
U25_GA(T206, X511, less71_out_g(T206)) → ADD102_IN_GA(T206, X509)
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U28_GA(T206, X511, f129_in_ga(T210, X510))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → U30_GA(T206, X511, f129_in_aa(T213, X511))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T185, X419) → U23_AA(T185, X419, less59_in_a(T185))
F129_IN_AA(T185, X419) → LESS59_IN_A(T185)
LESS59_IN_A(s(T93)) → U9_A(T93, less10_in_a(T93))
LESS59_IN_A(s(T93)) → LESS10_IN_A(T93)
LESS10_IN_A(s(T19)) → U2_A(T19, less16_in_a(T19))
LESS10_IN_A(s(T19)) → LESS16_IN_A(T19)
LESS16_IN_A(s(T22)) → U1_A(T22, less16_in_a(T22))
LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)
LESS10_IN_A(s(T25)) → U3_A(T25, less10_in_a(T25))
LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)
LESS59_IN_A(s(T97)) → U10_A(T97, less59_in_a(T97))
LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)
U23_AA(T185, X419, less59_out_a(T185)) → U24_AA(T185, X419, add136_in_ag(X419, T185))
U23_AA(T185, X419, less59_out_a(T185)) → ADD136_IN_AG(X419, T185)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
F129_IN_AA(T206, X511) → LESS71_IN_A(T206)
LESS71_IN_A(s(T116)) → U21_A(T116, less77_in_a(T116))
LESS71_IN_A(s(T116)) → LESS77_IN_A(T116)
LESS77_IN_A(s(T122)) → U11_A(T122, less83_in_a(T122))
LESS77_IN_A(s(T122)) → LESS83_IN_A(T122)
LESS77_IN_A(T141) → U12_A(T141, less83_in_a(T141))
LESS77_IN_A(T141) → LESS83_IN_A(T141)
LESS71_IN_A(T145) → U22_A(T145, less77_in_a(T145))
LESS71_IN_A(T145) → LESS77_IN_A(T145)
U25_AA(T206, X511, less71_out_a(T206)) → U26_AA(T206, X511, add102_in_ga(T206, X509))
U25_AA(T206, X511, less71_out_a(T206)) → ADD102_IN_GA(T206, X509)
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U28_AA(T206, X511, f129_in_ga(T210, X510))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → U30_AA(T206, X511, f129_in_aa(T213, X511))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
U37_GA(T107, T109, add102_out_ga(T107, T148)) → U39_GA(T107, T109, f129_in_ga(T148, T177))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → U40_GA(T107, T109, f1_in_aa(T177, T109))
U39_GA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(s(T12), T7) → U31_AA(T12, T7, less10_in_a(T12))
F1_IN_AA(s(T12), T7) → LESS10_IN_A(T12)
U31_AA(T12, T7, less10_out_a(T12)) → U32_AA(T12, T7, add11_in_ag(T7, T12))
U31_AA(T12, T7, less10_out_a(T12)) → ADD11_IN_AG(T7, T12)
F1_IN_AA(s(T86), T7) → U33_AA(T86, T7, less59_in_a(T86))
F1_IN_AA(s(T86), T7) → LESS59_IN_A(T86)
U33_AA(T86, T7, less59_out_a(T86)) → U34_AA(T86, T7, add11_in_ag(T7, T86))
U33_AA(T86, T7, less59_out_a(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
F1_IN_AA(T107, T109) → LESS71_IN_A(T107)
U35_AA(T107, T109, less71_out_a(T107)) → U36_AA(T107, T109, add102_in_ga(T107, X193))
U35_AA(T107, T109, less71_out_a(T107)) → ADD102_IN_GA(T107, X193)
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U38_AA(T107, T109, f129_in_ga(T148, X194))
U37_AA(T107, T109, add102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → U40_AA(T107, T109, f1_in_aa(T177, T109))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
LESS16_IN_G(x1)  =  LESS16_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_G(x1, x2)  =  U3_G(x1, x2)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
ADD11_IN_AG(x1, x2)  =  ADD11_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x2, x3)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
LESS71_IN_G(x1)  =  LESS71_IN_G(x1)
U21_G(x1, x2)  =  U21_G(x1, x2)
LESS77_IN_G(x1)  =  LESS77_IN_G(x1)
U11_G(x1, x2)  =  U11_G(x1, x2)
LESS83_IN_G(x1)  =  LESS83_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U22_G(x1, x2)  =  U22_G(x1, x2)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U37_GA(x1, x2, x3)  =  U37_GA(x1, x3)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)
U18_AG(x1, x2, x3)  =  U18_AG(x2, x3)
U19_AG(x1, x2, x3)  =  U19_AG(x2, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x2, x3)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U26_GA(x1, x2, x3)  =  U26_GA(x1, x3)
U27_GA(x1, x2, x3)  =  U27_GA(x1, x3)
U28_GA(x1, x2, x3)  =  U28_GA(x1, x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x1, x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U23_AA(x1, x2, x3)  =  U23_AA(x3)
LESS59_IN_A(x1)  =  LESS59_IN_A
U9_A(x1, x2)  =  U9_A(x2)
LESS10_IN_A(x1)  =  LESS10_IN_A
U2_A(x1, x2)  =  U2_A(x2)
LESS16_IN_A(x1)  =  LESS16_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U10_A(x1, x2)  =  U10_A(x2)
U24_AA(x1, x2, x3)  =  U24_AA(x1, x3)
U25_AA(x1, x2, x3)  =  U25_AA(x3)
LESS71_IN_A(x1)  =  LESS71_IN_A
U21_A(x1, x2)  =  U21_A(x2)
LESS77_IN_A(x1)  =  LESS77_IN_A
U11_A(x1, x2)  =  U11_A(x2)
LESS83_IN_A(x1)  =  LESS83_IN_A
U12_A(x1, x2)  =  U12_A(x2)
U22_A(x1, x2)  =  U22_A(x2)
U26_AA(x1, x2, x3)  =  U26_AA(x1, x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U28_AA(x1, x2, x3)  =  U28_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)
U30_AA(x1, x2, x3)  =  U30_AA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x1, x3)
U40_GA(x1, x2, x3)  =  U40_GA(x1, x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U31_AA(x1, x2, x3)  =  U31_AA(x3)
U32_AA(x1, x2, x3)  =  U32_AA(x1, x3)
U33_AA(x1, x2, x3)  =  U33_AA(x3)
U34_AA(x1, x2, x3)  =  U34_AA(x1, x3)
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U36_AA(x1, x2, x3)  =  U36_AA(x1, x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U38_AA(x1, x2, x3)  =  U38_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)
U40_AA(x1, x2, x3)  =  U40_AA(x1, x3)

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

(336) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 15 SCCs with 99 less nodes.

(337) Complex Obligation (AND)

(338) Obligation:

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

LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS16_IN_A(x1)  =  LESS16_IN_A

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

(339) UsableRulesProof (EQUIVALENT transformation)

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

(340) Obligation:

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

LESS16_IN_A(s(T22)) → LESS16_IN_A(T22)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS16_IN_A(x1)  =  LESS16_IN_A

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

(341) PiDPToQDPProof (SOUND transformation)

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

(342) Obligation:

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

LESS16_IN_A → LESS16_IN_A

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

(343) 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 = LESS16_IN_A evaluates to t =LESS16_IN_A

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 LESS16_IN_A to LESS16_IN_A.

(344) NO

(345) Obligation:

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

LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS10_IN_A(x1)  =  LESS10_IN_A

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

(346) UsableRulesProof (EQUIVALENT transformation)

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

(347) Obligation:

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

LESS10_IN_A(s(T25)) → LESS10_IN_A(T25)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS10_IN_A(x1)  =  LESS10_IN_A

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

(348) PiDPToQDPProof (SOUND transformation)

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

(349) Obligation:

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

LESS10_IN_A → LESS10_IN_A

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

(350) 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 = LESS10_IN_A evaluates to t =LESS10_IN_A

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 LESS10_IN_A to LESS10_IN_A.

(351) NO

(352) Obligation:

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

LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS59_IN_A(x1)  =  LESS59_IN_A

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

(353) UsableRulesProof (EQUIVALENT transformation)

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

(354) Obligation:

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

LESS59_IN_A(s(T97)) → LESS59_IN_A(T97)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS59_IN_A(x1)  =  LESS59_IN_A

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

(355) PiDPToQDPProof (SOUND transformation)

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

(356) Obligation:

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

LESS59_IN_A → LESS59_IN_A

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

(357) 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 = LESS59_IN_A evaluates to t =LESS59_IN_A

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 LESS59_IN_A to LESS59_IN_A.

(358) NO

(359) Obligation:

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

ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)

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

(360) UsableRulesProof (EQUIVALENT transformation)

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

(361) Obligation:

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

ADD143_IN_AG(s(X489), s(T202)) → ADD143_IN_AG(X489, T202)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)

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

(362) PiDPToQDPProof (SOUND transformation)

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

(363) Obligation:

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

ADD143_IN_AG(s(T202)) → ADD143_IN_AG(T202)

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

(364) 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:

• ADD143_IN_AG(s(T202)) → ADD143_IN_AG(T202)
  The graph contains the following edges 1 > 1

(365) YES

(366) Obligation:

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

ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)

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

(367) UsableRulesProof (EQUIVALENT transformation)

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

(368) Obligation:

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

ADD136_IN_AG(s(X447), s(T193)) → ADD136_IN_AG(X447, T193)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)

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

(369) PiDPToQDPProof (SOUND transformation)

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

(370) Obligation:

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

ADD136_IN_AG(s(T193)) → ADD136_IN_AG(T193)

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

(371) 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:

• ADD136_IN_AG(s(T193)) → ADD136_IN_AG(T193)
  The graph contains the following edges 1 > 1

(372) YES

(373) Obligation:

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

ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)

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

(374) UsableRulesProof (EQUIVALENT transformation)

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

(375) Obligation:

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

ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)

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

(376) PiDPToQDPProof (SOUND transformation)

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

(377) Obligation:

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

ADD117_IN_GA(s(T174)) → ADD117_IN_GA(T174)

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

(378) 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:

• ADD117_IN_GA(s(T174)) → ADD117_IN_GA(T174)
  The graph contains the following edges 1 > 1

(379) YES

(380) Obligation:

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

ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)

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

(381) UsableRulesProof (EQUIVALENT transformation)

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

(382) Obligation:

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

ADD110_IN_GA(s(T165), s(X349)) → ADD110_IN_GA(T165, X349)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)

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

(383) PiDPToQDPProof (SOUND transformation)

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

(384) Obligation:

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

ADD110_IN_GA(s(T165)) → ADD110_IN_GA(T165)

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

(385) 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:

• ADD110_IN_GA(s(T165)) → ADD110_IN_GA(T165)
  The graph contains the following edges 1 > 1

(386) YES

(387) Obligation:

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

ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)

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

(388) UsableRulesProof (EQUIVALENT transformation)

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

(389) Obligation:

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

ADD102_IN_GA(s(T154), s(X302)) → ADD102_IN_GA(T154, X302)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)

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

(390) PiDPToQDPProof (SOUND transformation)

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

(391) Obligation:

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

ADD102_IN_GA(s(T154)) → ADD102_IN_GA(T154)

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

(392) 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:

• ADD102_IN_GA(s(T154)) → ADD102_IN_GA(T154)
  The graph contains the following edges 1 > 1

(393) YES

(394) Obligation:

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

ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)

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

(395) UsableRulesProof (EQUIVALENT transformation)

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

(396) Obligation:

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

ADD41_IN_AG(s(T72), s(T71)) → ADD41_IN_AG(T72, T71)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)

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

(397) PiDPToQDPProof (SOUND transformation)

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

(398) Obligation:

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

ADD41_IN_AG(s(T71)) → ADD41_IN_AG(T71)

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

(399) 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:

• ADD41_IN_AG(s(T71)) → ADD41_IN_AG(T71)
  The graph contains the following edges 1 > 1

(400) YES

(401) Obligation:

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

ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)

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

(402) UsableRulesProof (EQUIVALENT transformation)

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

(403) Obligation:

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

ADD33_IN_AG(s(T52), s(T51)) → ADD33_IN_AG(T52, T51)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)

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

(404) PiDPToQDPProof (SOUND transformation)

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

(405) Obligation:

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

ADD33_IN_AG(s(T51)) → ADD33_IN_AG(T51)

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

(406) 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:

• ADD33_IN_AG(s(T51)) → ADD33_IN_AG(T51)
  The graph contains the following edges 1 > 1

(407) YES

(408) Obligation:

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

LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS16_IN_G(x1)  =  LESS16_IN_G(x1)

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

(409) UsableRulesProof (EQUIVALENT transformation)

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

(410) Obligation:

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

LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)

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

(411) PiDPToQDPProof (EQUIVALENT transformation)

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

(412) Obligation:

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

LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)

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

(413) 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:

• LESS16_IN_G(s(T22)) → LESS16_IN_G(T22)
  The graph contains the following edges 1 > 1

(414) YES

(415) Obligation:

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

LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)

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

(416) UsableRulesProof (EQUIVALENT transformation)

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

(417) Obligation:

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

LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)

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

(418) PiDPToQDPProof (EQUIVALENT transformation)

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

(419) Obligation:

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

LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)

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

(420) 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:

• LESS10_IN_G(s(T25)) → LESS10_IN_G(T25)
  The graph contains the following edges 1 > 1

(421) YES

(422) Obligation:

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

LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)

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

(423) UsableRulesProof (EQUIVALENT transformation)

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

(424) Obligation:

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

LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)

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

(425) PiDPToQDPProof (EQUIVALENT transformation)

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

(426) Obligation:

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

LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)

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

(427) 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:

• LESS59_IN_G(s(T97)) → LESS59_IN_G(T97)
  The graph contains the following edges 1 > 1

(428) YES

(429) Obligation:

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

F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U27_GA(x1, x2, x3)  =  U27_GA(x1, x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U25_AA(x1, x2, x3)  =  U25_AA(x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)

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

(430) UsableRulesProof (EQUIVALENT transformation)

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

(431) Obligation:

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

F129_IN_GA(T206, X511) → U25_GA(T206, X511, less71_in_g(T206))
U25_GA(T206, X511, less71_out_g(T206)) → U27_GA(T206, X511, add102_in_ga(T206, T210))
U27_GA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_GA(T206, X511, add102_out_ga(T206, T210)) → U29_GA(T206, X511, f129_in_ga(T210, T213))
U29_GA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)
F129_IN_AA(T206, X511) → U25_AA(T206, X511, less71_in_a(T206))
U25_AA(T206, X511, less71_out_a(T206)) → U27_AA(T206, X511, add102_in_ga(T206, T210))
U27_AA(T206, X511, add102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U27_AA(T206, X511, add102_out_ga(T206, T210)) → U29_AA(T206, X511, f129_in_ga(T210, T213))
U29_AA(T206, X511, f129_out_ga(T210, T213)) → F129_IN_AA(T213, X511)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
less83_in_a(0) → less83_out_a(0)
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
U27_GA(x1, x2, x3)  =  U27_GA(x1, x3)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
F129_IN_AA(x1, x2)  =  F129_IN_AA
U25_AA(x1, x2, x3)  =  U25_AA(x3)
U27_AA(x1, x2, x3)  =  U27_AA(x1, x3)
U29_AA(x1, x2, x3)  =  U29_AA(x1, x3)

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

(432) PiDPToQDPProof (SOUND transformation)

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

(433) Obligation:

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

F129_IN_GA(T206) → U25_GA(T206, less71_in_g(T206))
U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_in_a)
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(434) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(T206) → U25_GA(T206, less71_in_g(T206)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, less77_in_g(x0)))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, less77_in_g(x0)))

(435) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_in_a)
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, less77_in_g(x0)))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, less77_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(436) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_AA → U25_AA(less71_in_a) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))

(437) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, less77_in_g(x0)))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, less77_in_g(x0)))
F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(438) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, less77_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))

(439) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, less77_in_g(x0)))
F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(440) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, less77_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))

(441) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U21_a(less77_in_a))
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(442) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(less77_in_a)) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a)))

(443) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(444) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U11_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0))))

(445) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a)))
F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(446) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U12_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0))))

(447) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(448) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U11_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0))))

(449) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(450) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(U12_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U21_a(less77_out_a(0)))

(451) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(452) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(less77_out_a(s(0)))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(s(s(0))))

(453) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(U21_a(less77_out_a(0)))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(454) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U21_a(less77_out_a(0))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(s(0)))

(455) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(U22_a(less77_in_a))
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(456) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(less77_in_a)) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a)))

(457) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U11_a(less83_in_a)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(458) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U11_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0))))

(459) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a)))
F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(460) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U12_a(less83_in_a))) at position [0,0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0))))

(461) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(462) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U11_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0))))

(463) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0))))
F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(464) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(U12_a(less83_out_a(0)))) at position [0,0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(U22_a(less77_out_a(0)))

(465) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0))))
F129_IN_AA → U25_AA(U22_a(less77_out_a(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(466) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(less77_out_a(s(0)))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(s(0)))

(467) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(U22_a(less77_out_a(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(468) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_AA → U25_AA(U22_a(less77_out_a(0))) at position [0] we obtained the following new rules [LPAR04]:

F129_IN_AA → U25_AA(less71_out_a(0))

(469) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(470) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(s(x0))) → U25_GA(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))

(471) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(472) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(s(0), less77_out_g(s(0))))

(473) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(s(0), less77_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(474) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), U21_g(s(0), less77_out_g(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))

(475) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(476) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U25_GA(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(0, U12_g(0, less83_out_g(0))))

(477) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(0, U12_g(0, less83_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(478) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(0, U12_g(0, less83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(0, less77_out_g(0)))

(479) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(0, less77_out_g(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(480) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U21_g(0, less77_out_g(0))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))

(481) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(482) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U25_GA(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(s(0), U11_g(0, less83_out_g(0))))

(483) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(s(0), U11_g(0, less83_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(484) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(s(0), less77_out_g(s(0))))

(485) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(s(0), less77_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(486) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U25_GA(s(0), U22_g(s(0), less77_out_g(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))

(487) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(488) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(x0) → U25_GA(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U25_GA(0, U22_g(0, U12_g(0, less83_out_g(0))))

(489) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, U22_g(0, U12_g(0, less83_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(490) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(0) → U25_GA(0, U22_g(0, U12_g(0, less83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U25_GA(0, U22_g(0, less77_out_g(0)))

(491) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, U22_g(0, less77_out_g(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(492) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(0) → U25_GA(0, U22_g(0, less77_out_g(0))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U25_GA(0, less71_out_g(0))

(493) Obligation:

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

U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206))
U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(494) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_GA(T206, less71_out_g(T206)) → U27_GA(T206, add102_in_ga(T206)) we obtained the following new rules [LPAR04]:

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))

(495) Obligation:

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

U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(496) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_GA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210) we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)

(497) Obligation:

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

U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210))
U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(498) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_GA(T206, add102_out_ga(T206, T210)) → U29_GA(T206, f129_in_ga(T210)) we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))

(499) Obligation:

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

U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA
U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(500) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U29_GA(T206, f129_out_ga(T210)) → F129_IN_AA we obtained the following new rules [LPAR04]:

U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA

(501) Obligation:

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

U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206))
U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(502) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_AA(less71_out_a(T206)) → U27_AA(T206, add102_in_ga(T206)) we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))

(503) Obligation:

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

U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210)
U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(504) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(T206, add102_out_ga(T206, T210)) → F129_IN_GA(T210) we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)

(505) Obligation:

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

U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210))
U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(506) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(T206, add102_out_ga(T206, T210)) → U29_AA(T206, f129_in_ga(T210)) we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))

(507) Obligation:

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

U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(508) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U29_AA(T206, f129_out_ga(T210)) → F129_IN_AA we obtained the following new rules [LPAR04]:

U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA

(509) Obligation:

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

F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(510) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U25_GA(0, less71_out_g(0)) → U27_GA(0, add102_in_ga(0))

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

POL(0) = 1   
POL(F129_IN_AA) = 0   
POL(F129_IN_GA(x₁)) = x₁   
POL(U10_a(x₁)) = 0   
POL(U10_g(x₁, x₂)) = 0   
POL(U11_a(x₁)) = x₁   
POL(U11_g(x₁, x₂)) = 0   
POL(U12_a(x₁)) = 1   
POL(U12_g(x₁, x₂)) = 0   
POL(U13_ga(x₁, x₂)) = 0   
POL(U14_ga(x₁, x₂)) = 0   
POL(U15_ga(x₁, x₂)) = 0   
POL(U16_ga(x₁, x₂)) = 0   
POL(U17_ga(x₁, x₂)) = 0   
POL(U18_ag(x₁, x₂)) = 0   
POL(U19_ag(x₁, x₂)) = 0   
POL(U1_a(x₁)) = 0   
POL(U1_g(x₁, x₂)) = 0   
POL(U20_ag(x₁, x₂)) = 0   
POL(U21_a(x₁)) = 1   
POL(U21_g(x₁, x₂)) = 0   
POL(U22_a(x₁)) = 1   
POL(U22_g(x₁, x₂)) = 0   
POL(U23_aa(x₁)) = 0   
POL(U23_ga(x₁, x₂)) = 0   
POL(U24_aa(x₁, x₂)) = 0   
POL(U24_ga(x₁, x₂)) = 0   
POL(U25_AA(x₁)) = 0   
POL(U25_GA(x₁, x₂)) = x₁   
POL(U25_aa(x₁)) = 0   
POL(U25_ga(x₁, x₂)) = 0   
POL(U26_aa(x₁, x₂)) = 0   
POL(U26_ga(x₁, x₂)) = 0   
POL(U27_AA(x₁, x₂)) = x₂   
POL(U27_GA(x₁, x₂)) = x₂   
POL(U27_aa(x₁, x₂)) = 0   
POL(U27_ga(x₁, x₂)) = 0   
POL(U28_aa(x₁, x₂)) = 0   
POL(U28_ga(x₁, x₂)) = 0   
POL(U29_AA(x₁, x₂)) = 0   
POL(U29_GA(x₁, x₂)) = 0   
POL(U29_aa(x₁, x₂)) = 0   
POL(U29_ga(x₁, x₂)) = 0   
POL(U2_a(x₁)) = 0   
POL(U2_g(x₁, x₂)) = 0   
POL(U30_aa(x₁, x₂)) = 0   
POL(U30_ga(x₁, x₂)) = 0   
POL(U3_a(x₁)) = 0   
POL(U3_g(x₁, x₂)) = 0   
POL(U9_a(x₁)) = 0   
POL(U9_g(x₁, x₂)) = 0   
POL(add102_in_ga(x₁)) = 0   
POL(add102_out_ga(x₁, x₂)) = x₂   
POL(add110_in_ga(x₁)) = 0   
POL(add110_out_ga(x₁, x₂)) = 0   
POL(add117_in_ga(x₁)) = 0   
POL(add117_out_ga(x₁, x₂)) = 0   
POL(add136_in_ag(x₁)) = 0   
POL(add136_out_ag(x₁, x₂)) = 0   
POL(add143_in_ag(x₁)) = 0   
POL(add143_out_ag(x₁, x₂)) = 0   
POL(f129_in_aa) = 0   
POL(f129_in_ga(x₁)) = 0   
POL(f129_out_aa(x₁)) = 0   
POL(f129_out_ga(x₁)) = 0   
POL(less10_in_a) = 0   
POL(less10_in_g(x₁)) = 1   
POL(less10_out_a(x₁)) = 0   
POL(less10_out_g(x₁)) = 0   
POL(less16_in_a) = 0   
POL(less16_in_g(x₁)) = 0   
POL(less16_out_a(x₁)) = 0   
POL(less16_out_g(x₁)) = 0   
POL(less59_in_a) = 0   
POL(less59_in_g(x₁)) = 0   
POL(less59_out_a(x₁)) = 0   
POL(less59_out_g(x₁)) = 0   
POL(less71_in_a) = 1   
POL(less71_in_g(x₁)) = 0   
POL(less71_out_a(x₁)) = 1   
POL(less71_out_g(x₁)) = 0   
POL(less77_in_a) = 1   
POL(less77_in_g(x₁)) = 0   
POL(less77_out_a(x₁)) = 0   
POL(less77_out_g(x₁)) = 0   
POL(less83_in_a) = 1   
POL(less83_in_g(x₁)) = 0   
POL(less83_out_a(x₁)) = 0   
POL(less83_out_g(x₁)) = 0   
POL(s(x₁)) = 0   

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))

(511) Obligation:

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

F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
F129_IN_AA → U25_AA(less71_out_a(0))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
F129_IN_GA(0) → U25_GA(0, less71_out_g(0))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U27_GA(0, add102_out_ga(0, x1)) → U29_GA(0, f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_GA(0, f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(512) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(513) Obligation:

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

U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(514) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), add102_in_ga(s(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

(515) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(516) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), add102_in_ga(s(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

(517) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(518) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), add102_in_ga(s(0))) at position [1] we obtained the following new rules [LPAR04]:

U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))

(519) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(520) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), x1)) → U29_GA(s(0), f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))

(521) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(522) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_AA(less71_out_a(s(0))) → U27_AA(s(0), add102_in_ga(s(0))) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))

(523) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(524) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), x1)) → U29_AA(s(0), f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))

(525) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(526) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, add102_in_ga(0)) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, add110_in_ga(0)))

(527) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, add110_in_ga(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(528) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, add110_in_ga(0))) at position [1,1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, U16_ga(0, add117_in_ga(0))))

(529) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, U16_ga(0, add117_in_ga(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(530) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, U16_ga(0, add117_in_ga(0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, U16_ga(0, add117_out_ga(0, 0))))

(531) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, U16_ga(0, add117_out_ga(0, 0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(532) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, U16_ga(0, add117_out_ga(0, 0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, add110_out_ga(0, s(0))))

(533) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, add110_out_ga(0, s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(534) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U25_AA(less71_out_a(0)) → U27_AA(0, U14_ga(0, add110_out_ga(0, s(0)))) at position [1] we obtained the following new rules [LPAR04]:

U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))

(535) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(536) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, x1)) → U29_AA(0, f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))

(537) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(538) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_GA(s(s(0)), f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0)))

(539) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(540) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → U29_AA(s(s(0)), f129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))

(541) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(542) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

(543) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(544) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

(545) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(546) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))

(547) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(548) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

(549) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(550) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, less71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

(551) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(552) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

(553) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(554) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

(555) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(556) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

(557) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(558) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

(559) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(560) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

(561) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(562) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

(563) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(564) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

(565) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(566) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

(567) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(568) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

(569) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(570) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, less77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

(571) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(572) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_GA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

(573) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(574) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

(575) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(576) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

(577) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(578) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

(579) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(580) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U21_g(0, less77_out_g(0))))

(581) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U21_g(0, less77_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(582) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U21_g(0, less77_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))

(583) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(584) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(x0))) → U29_GA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

(585) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(586) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

(587) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(588) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))

(589) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(590) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), 0)) → U29_GA(s(0), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

(591) Obligation:

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

U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(0), add102_out_ga(s(0), 0)) → U29_GA(s(0), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(592) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(593) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(594) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, s(s(x0)))) → U29_AA(0, U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

(595) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(596) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

(597) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(598) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))

(599) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(600) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(0))) → U29_AA(0, U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

(601) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(0))) → U29_AA(0, U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(602) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(603) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(604) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, s(x0))) → U29_AA(0, U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(0))) → U29_AA(0, U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

(605) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(0))) → U29_AA(0, U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(606) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(607) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(608) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, 0)) → U29_AA(0, U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

(609) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, 0)) → U29_AA(0, U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(610) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(611) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(612) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(s(x0)))) → U29_AA(s(0), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

(613) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(614) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

(615) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(616) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

(617) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(618) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

(619) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(620) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(0, less77_out_g(0))))

(621) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(0, less77_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(622) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U21_g(0, less77_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))

(623) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(624) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(x0))) → U29_AA(s(0), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

(625) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(626) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

(627) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(628) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))

(629) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(630) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), 0)) → U29_AA(s(0), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

(631) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(0), add102_out_ga(s(0), 0)) → U29_AA(s(0), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(632) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(633) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(634) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_AA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

(635) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(636) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

(637) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(638) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

(639) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(640) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

(641) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(642) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(0, less77_out_g(0))))

(643) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(0, less77_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(644) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U21_g(0, less77_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

(645) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(646) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_AA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

(647) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(648) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

(649) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(650) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

(651) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(652) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), 0)) → U29_AA(s(s(0)), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

(653) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), 0)) → U29_AA(s(s(0)), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(654) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(655) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(656) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(x0)))) → U29_GA(s(s(0)), U25_ga(s(s(x0)), U21_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

(657) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(658) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

(659) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(660) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), U21_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

(661) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(662) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U21_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

(663) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(664) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U21_g(0, U12_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U21_g(0, less77_out_g(0))))

(665) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U21_g(0, less77_out_g(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(666) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U21_g(0, less77_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

(667) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(668) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(x0))) → U29_GA(s(s(0)), U25_ga(s(x0), U22_g(s(x0), U11_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

(669) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(670) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U22_g(s(0), U11_g(0, less83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

(671) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(672) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), U22_g(s(0), less77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

(673) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(674) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U25_ga(x0, U22_g(x0, U12_g(x0, less83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), 0)) → U29_GA(s(s(0)), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

(675) Obligation:

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

U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), 0)) → U29_GA(s(s(0)), U25_ga(0, U22_g(0, U12_g(0, less83_out_g(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(676) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(677) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1)
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(678) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(0, add102_out_ga(0, x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))

(679) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0)))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(680) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_AA(0, add102_out_ga(0, x0)) → U29_AA(0, U23_ga(x0, less59_in_g(x0))) we obtained the following new rules [LPAR04]:

U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))

(681) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(682) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_GA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))

(683) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(684) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_GA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

(685) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(686) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_AA(s(s(0)), add102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))

(687) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(688) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U27_AA(s(0), add102_out_ga(s(0), x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

(689) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(690) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U27_AA(s(0), add102_out_ga(s(0), s(0))) → U29_AA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_AA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

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

POL(F129_IN_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(0) =

/ 1 \ \ 1 /

POL(U25_GA(x1, x2)) =

0

+

[ 1, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(less71_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U27_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 1 ]

·

x2

POL(U13_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(add102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(add102_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U29_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(f129_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(F129_IN_AA) =

0

POL(U25_AA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(add110_in_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(less10_in_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U10_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U24_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add136_in_ag(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U26_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U15_ga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U16_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add117_in_ga(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(add110_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U28_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U21_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less77_in_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U22_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(U29_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U30_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_aa) =

/ 1 \ \ 1 /

POL(U23_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 1 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 0 /

POL(f129_out_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 1 \ \ 0 /

POL(U21_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 1 \ \ 1 /

POL(U22_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U30_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U11_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less83_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U12_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(less77_out_g(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U11_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(less83_in_a) =

/ 1 \ \ 1 /

POL(U12_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less77_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U18_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U19_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_in_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(add136_out_ag(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(less83_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(add117_out_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U17_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(U2_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less16_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U3_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less10_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less83_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U9_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less10_in_a) =

/ 1 \ \ 1 /

POL(U10_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less59_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U24_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(add143_out_ag(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U20_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less16_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less16_in_a) =

/ 0 \ \ 1 /

POL(U3_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less16_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
f129_in_aa → U25_aa(less71_in_a)
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))

(691) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(692) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

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

POL(F129_IN_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(0) =

/ 1 \ \ 0 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(less71_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(U13_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(add102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add102_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 1 0 /

·

x2

POL(U29_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(f129_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(F129_IN_AA) =

0

POL(U25_AA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(add110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less10_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U10_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U24_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add136_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U15_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U16_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add110_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(U28_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 1 /

·

x1

POL(less71_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U21_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(less77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U22_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U30_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_aa) =

/ 0 \ \ 0 /

POL(U23_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 1 /

POL(f129_out_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 1 \ \ 0 /

POL(U21_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 0 \ \ 1 /

POL(U22_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U30_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U11_g(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(less83_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(U12_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less77_out_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U11_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(less83_in_a) =

/ 0 \ \ 0 /

POL(U12_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(U18_ag(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U19_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add136_out_ag(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(less83_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(add117_out_ga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

POL(U17_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_g(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less16_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U3_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(less10_out_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less83_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U9_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less10_in_a) =

/ 1 \ \ 0 /

POL(U10_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less59_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U24_aa(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_out_ag(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U20_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(less16_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less16_in_a) =

/ 0 \ \ 1 /

POL(U3_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less16_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
f129_in_aa → U25_aa(less71_in_a)
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))

(693) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(694) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U27_GA(s(0), add102_out_ga(s(0), s(0))) → U29_GA(s(0), U25_ga(s(0), less71_out_g(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → U29_GA(s(s(0)), U25_ga(s(0), less71_out_g(s(0))))

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

POL(F129_IN_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(0) =

/ 0 \ \ 1 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(less71_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U13_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(add102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add102_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U29_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(f129_out_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(F129_IN_AA) =

0

POL(U25_AA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(add110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less10_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U10_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(less59_out_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U24_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add136_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U26_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_ga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U15_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U16_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add110_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U28_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_ga(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(less71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U21_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(less77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U22_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U29_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U30_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_aa) =

/ 0 \ \ 1 /

POL(U23_aa(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 0 /

POL(f129_out_aa(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 0 \ \ 0 /

POL(U21_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 0 \ \ 0 /

POL(U22_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 1 0 \ \ 1 1 /

·

x2

POL(U30_aa(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U11_g(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less83_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U12_g(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(less77_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U11_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less83_in_a) =

/ 0 \ \ 0 /

POL(U12_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(less77_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U18_ag(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 1 0 /

·

x2

POL(U19_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_in_ag(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL(add136_out_ag(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less83_out_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(add117_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U17_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(less16_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U3_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less10_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less83_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U9_a(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less10_in_a) =

/ 0 \ \ 0 /

POL(U10_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U24_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_out_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U20_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(less16_out_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(less16_in_a) =

/ 1 \ \ 1 /

POL(U3_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_out_a(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less16_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))

(695) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(696) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U27_AA(s(0), add102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

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

POL(F129_IN_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(0) =

/ 0 \ \ 1 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(less71_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U13_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(add102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add102_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U29_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U23_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(f129_out_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(F129_IN_AA) =

0

POL(U25_AA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(less71_out_a(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U29_AA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U25_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(add110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less10_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U10_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(less59_out_g(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U24_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(add136_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U26_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_ga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U15_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U16_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add110_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U28_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(f129_in_ga(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U21_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(less77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U22_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U29_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U30_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(f129_in_aa) =

/ 1 \ \ 1 /

POL(U23_aa(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(less59_in_a) =

/ 0 \ \ 0 /

POL(f129_out_aa(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL(U25_aa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less71_in_a) =

/ 1 \ \ 1 /

POL(U21_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less77_in_a) =

/ 0 \ \ 0 /

POL(U22_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U26_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U27_aa(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U28_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U29_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U30_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U11_g(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(less83_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U12_g(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less77_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U11_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less83_in_a) =

/ 0 \ \ 1 /

POL(U12_a(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(less77_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(U18_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U19_ag(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(add143_in_ag(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add136_out_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less83_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(add117_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U17_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less16_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U3_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less10_out_g(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(less83_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less10_in_a) =

/ 0 \ \ 0 /

POL(U10_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(less59_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U24_aa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(add143_out_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U20_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(less16_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U2_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less16_in_a) =

/ 0 \ \ 0 /

POL(U3_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(less10_out_a(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(less16_out_a(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U1_a(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))

(697) Obligation:

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

F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
F129_IN_GA(s(0)) → U25_GA(s(0), less71_out_g(s(0)))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U13_ga(0, add102_in_ga(0)))
U27_GA(s(0), add102_out_ga(s(0), x0)) → U29_GA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_GA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_GA(s(0), less71_out_g(s(0))) → U27_GA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_GA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(698) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(699) Obligation:

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

U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_GA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_GA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(s(0))))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U13_ga(s(0), add102_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), x0)) → U29_AA(s(s(0)), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(s(0)), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U13_ga(0, add102_in_ga(0)))
U27_AA(s(0), add102_out_ga(s(0), x0)) → U29_AA(s(0), U23_ga(x0, less59_in_g(x0)))
U29_AA(s(0), f129_out_ga(x1)) → F129_IN_AA
F129_IN_AA → U25_AA(less71_out_a(0))
U25_AA(less71_out_a(0)) → U27_AA(0, add102_out_ga(0, s(s(0))))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U29_AA(0, f129_out_ga(x1)) → F129_IN_AA
U27_AA(0, add102_out_ga(0, s(s(0)))) → F129_IN_GA(s(s(0)))
F129_IN_GA(s(s(0))) → U25_GA(s(s(0)), less71_out_g(s(s(0))))
U25_GA(s(s(0)), less71_out_g(s(s(0)))) → U27_GA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_GA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_GA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U27_AA(0, add102_out_ga(0, s(s(0)))) → U29_AA(0, U23_ga(s(s(0)), less59_in_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → U29_AA(s(0), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(0), add102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U25_AA(less71_out_a(s(0))) → U27_AA(s(0), U14_ga(s(0), add110_in_ga(s(0))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → U29_AA(s(s(0)), U25_ga(s(s(0)), less71_out_g(s(s(0)))))
U27_AA(s(s(0)), add102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U25_AA(less71_out_a(s(s(0)))) → U27_AA(s(s(0)), U14_ga(s(s(0)), add110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
add102_in_ga(s(T154)) → U13_ga(T154, add102_in_ga(T154))
add102_in_ga(T159) → U14_ga(T159, add110_in_ga(T159))
f129_in_ga(T185) → U23_ga(T185, less59_in_g(T185))
f129_in_ga(T206) → U25_ga(T206, less71_in_g(T206))
less71_in_a → U21_a(less77_in_a)
less71_in_a → U22_a(less77_in_a)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U13_ga(T154, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U23_ga(T185, less59_out_g(T185)) → f129_out_ga(T185)
U23_ga(T185, less59_out_g(T185)) → U24_ga(T185, add136_in_ag(T185))
U25_ga(T206, less71_out_g(T206)) → f129_out_ga(T206)
U25_ga(T206, less71_out_g(T206)) → U26_ga(T206, add102_in_ga(T206))
U25_ga(T206, less71_out_g(T206)) → U27_ga(T206, add102_in_ga(T206))
U21_a(less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(less77_out_a(T145)) → less71_out_a(T145)
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
add110_in_ga(s(T165)) → U15_ga(T165, add110_in_ga(T165))
add110_in_ga(T169) → U16_ga(T169, add117_in_ga(T169))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, add136_out_ag(X419, T185)) → f129_out_ga(T185)
U26_ga(T206, add102_out_ga(T206, X509)) → f129_out_ga(T206)
U27_ga(T206, add102_out_ga(T206, T210)) → U28_ga(T206, f129_in_ga(T210))
U27_ga(T206, add102_out_ga(T206, T210)) → U29_ga(T206, f129_in_ga(T210))
less77_in_a → U11_a(less83_in_a)
less77_in_a → U12_a(less83_in_a)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U15_ga(T165, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(T193)) → U18_ag(T193, add136_in_ag(T193))
add136_in_ag(s(T197)) → U19_ag(T197, add143_in_ag(T197))
U28_ga(T206, f129_out_ga(T210)) → f129_out_ga(T206)
U29_ga(T206, f129_out_ga(T210)) → U30_ga(T206, f129_in_aa)
U11_a(less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(less83_out_a(T141)) → less77_out_a(T141)
less83_in_g(0) → less83_out_g(0)
add117_in_ga(0) → add117_out_ga(0, 0)
add117_in_ga(s(T174)) → U17_ga(T174, add117_in_ga(T174))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U30_ga(T206, f129_out_aa(T213)) → f129_out_ga(T206)
less83_in_a → less83_out_a(0)
U17_ga(T174, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0) → add143_out_ag(0, 0)
add143_in_ag(s(T202)) → U20_ag(T202, add143_in_ag(T202))
f129_in_aa → U23_aa(less59_in_a)
f129_in_aa → U25_aa(less71_in_a)
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U23_aa(less59_out_a(T185)) → f129_out_aa(T185)
U23_aa(less59_out_a(T185)) → U24_aa(T185, add136_in_ag(T185))
U25_aa(less71_out_a(T206)) → f129_out_aa(T206)
U25_aa(less71_out_a(T206)) → U26_aa(T206, add102_in_ga(T206))
U25_aa(less71_out_a(T206)) → U27_aa(T206, add102_in_ga(T206))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a → U9_a(less10_in_a)
less59_in_a → U10_a(less59_in_a)
U24_aa(T185, add136_out_ag(X419, T185)) → f129_out_aa(T185)
U26_aa(T206, add102_out_ga(T206, X509)) → f129_out_aa(T206)
U27_aa(T206, add102_out_ga(T206, T210)) → U28_aa(T206, f129_in_ga(T210))
U27_aa(T206, add102_out_ga(T206, T210)) → U29_aa(T206, f129_in_ga(T210))
U9_a(less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, f129_out_ga(T210)) → f129_out_aa(T206)
U29_aa(T206, f129_out_ga(T210)) → U30_aa(T206, f129_in_aa)
less10_in_a → U2_a(less16_in_a)
less10_in_a → U3_a(less10_in_a)
U30_aa(T206, f129_out_aa(T213)) → f129_out_aa(T206)
U2_a(less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a → less16_out_a(s(0))
less16_in_a → U1_a(less16_in_a)
U1_a(less16_out_a(T22)) → less16_out_a(s(T22))

The set Q consists of the following terms:

less71_in_g(x0)
add102_in_ga(x0)
f129_in_ga(x0)
less71_in_a
U21_g(x0, x1)
U22_g(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)
U23_ga(x0, x1)
U25_ga(x0, x1)
U21_a(x0)
U22_a(x0)
less77_in_g(x0)
add110_in_ga(x0)
less59_in_g(x0)
U24_ga(x0, x1)
U26_ga(x0, x1)
U27_ga(x0, x1)
less77_in_a
U11_g(x0, x1)
U12_g(x0, x1)
U15_ga(x0, x1)
U16_ga(x0, x1)
U9_g(x0, x1)
U10_g(x0, x1)
add136_in_ag(x0)
U28_ga(x0, x1)
U29_ga(x0, x1)
U11_a(x0)
U12_a(x0)
less83_in_g(x0)
add117_in_ga(x0)
less10_in_g(x0)
U18_ag(x0, x1)
U19_ag(x0, x1)
U30_ga(x0, x1)
less83_in_a
U17_ga(x0, x1)
U2_g(x0, x1)
U3_g(x0, x1)
add143_in_ag(x0)
f129_in_aa
less16_in_g(x0)
U20_ag(x0, x1)
U23_aa(x0)
U25_aa(x0)
U1_g(x0, x1)
less59_in_a
U24_aa(x0, x1)
U26_aa(x0, x1)
U27_aa(x0, x1)
U9_a(x0)
U10_a(x0)
U28_aa(x0, x1)
U29_aa(x0, x1)
less10_in_a
U30_aa(x0, x1)
U2_a(x0)
U3_a(x0)
less16_in_a
U1_a(x0)

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

(700) Obligation:

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

U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))

The TRS R consists of the following rules:

f1_in_ga(s(T12), T7) → U31_ga(T12, T7, less10_in_g(T12))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
U31_ga(T12, T7, less10_out_g(T12)) → f1_out_ga(s(T12), T7)
U31_ga(T12, T7, less10_out_g(T12)) → U32_ga(T12, T7, add11_in_ag(T7, T12))
add11_in_ag(s(T39), T38) → U7_ag(T39, T38, add33_in_ag(T39, T38))
add33_in_ag(s(T52), s(T51)) → U4_ag(T52, T51, add33_in_ag(T52, T51))
add33_in_ag(T61, s(T60)) → U5_ag(T61, T60, add41_in_ag(T61, T60))
add41_in_ag(0, 0) → add41_out_ag(0, 0)
add41_in_ag(s(T72), s(T71)) → U6_ag(T72, T71, add41_in_ag(T72, T71))
U6_ag(T72, T71, add41_out_ag(T72, T71)) → add41_out_ag(s(T72), s(T71))
U5_ag(T61, T60, add41_out_ag(T61, T60)) → add33_out_ag(T61, s(T60))
U4_ag(T52, T51, add33_out_ag(T52, T51)) → add33_out_ag(s(T52), s(T51))
U7_ag(T39, T38, add33_out_ag(T39, T38)) → add11_out_ag(s(T39), T38)
add11_in_ag(T81, T80) → U8_ag(T81, T80, add41_in_ag(T81, T80))
U8_ag(T81, T80, add41_out_ag(T81, T80)) → add11_out_ag(T81, T80)
U32_ga(T12, T7, add11_out_ag(T7, T12)) → f1_out_ga(s(T12), T7)
f1_in_ga(s(T86), T7) → U33_ga(T86, T7, less59_in_g(T86))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
U33_ga(T86, T7, less59_out_g(T86)) → f1_out_ga(s(T86), T7)
U33_ga(T86, T7, less59_out_g(T86)) → U34_ga(T86, T7, add11_in_ag(T7, T86))
U34_ga(T86, T7, add11_out_ag(T7, T86)) → f1_out_ga(s(T86), T7)
f1_in_ga(T107, T109) → U35_ga(T107, T109, less71_in_g(T107))
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less83_in_g(0) → less83_out_g(0)
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U35_ga(T107, T109, less71_out_g(T107)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U36_ga(T107, T109, add102_in_ga(T107, X193))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U36_ga(T107, T109, add102_out_ga(T107, X193)) → f1_out_ga(T107, T109)
U35_ga(T107, T109, less71_out_g(T107)) → U37_ga(T107, T109, add102_in_ga(T107, T148))
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U38_ga(T107, T109, f129_in_ga(T148, X194))
f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less83_in_a(0) → less83_out_a(0)
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
U38_ga(T107, T109, f129_out_ga(T148, X194)) → f1_out_ga(T107, T109)
U37_ga(T107, T109, add102_out_ga(T107, T148)) → U39_ga(T107, T109, f129_in_ga(T148, T177))
U39_ga(T107, T109, f129_out_ga(T148, T177)) → U40_ga(T107, T109, f1_in_aa(T177, T109))
f1_in_aa(s(T12), T7) → U31_aa(T12, T7, less10_in_a(T12))
U31_aa(T12, T7, less10_out_a(T12)) → f1_out_aa(s(T12), T7)
U31_aa(T12, T7, less10_out_a(T12)) → U32_aa(T12, T7, add11_in_ag(T7, T12))
U32_aa(T12, T7, add11_out_ag(T7, T12)) → f1_out_aa(s(T12), T7)
f1_in_aa(s(T86), T7) → U33_aa(T86, T7, less59_in_a(T86))
U33_aa(T86, T7, less59_out_a(T86)) → f1_out_aa(s(T86), T7)
U33_aa(T86, T7, less59_out_a(T86)) → U34_aa(T86, T7, add11_in_ag(T7, T86))
U34_aa(T86, T7, add11_out_ag(T7, T86)) → f1_out_aa(s(T86), T7)
f1_in_aa(T107, T109) → U35_aa(T107, T109, less71_in_a(T107))
U35_aa(T107, T109, less71_out_a(T107)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U36_aa(T107, T109, add102_in_ga(T107, X193))
U36_aa(T107, T109, add102_out_ga(T107, X193)) → f1_out_aa(T107, T109)
U35_aa(T107, T109, less71_out_a(T107)) → U37_aa(T107, T109, add102_in_ga(T107, T148))
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U38_aa(T107, T109, f129_in_ga(T148, X194))
U38_aa(T107, T109, f129_out_ga(T148, X194)) → f1_out_aa(T107, T109)
U37_aa(T107, T109, add102_out_ga(T107, T148)) → U39_aa(T107, T109, f129_in_ga(T148, T177))
U39_aa(T107, T109, f129_out_ga(T148, T177)) → U40_aa(T107, T109, f1_in_aa(T177, T109))
U40_aa(T107, T109, f1_out_aa(T177, T109)) → f1_out_aa(T107, T109)
U40_ga(T107, T109, f1_out_aa(T177, T109)) → f1_out_ga(T107, T109)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
U31_ga(x1, x2, x3)  =  U31_ga(x1, x3)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f1_out_ga(x1, x2)  =  f1_out_ga(x1)
U32_ga(x1, x2, x3)  =  U32_ga(x1, x3)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x2, x3)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
add41_out_ag(x1, x2)  =  add41_out_ag(x1, x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
add33_out_ag(x1, x2)  =  add33_out_ag(x1, x2)
add11_out_ag(x1, x2)  =  add11_out_ag(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
U33_ga(x1, x2, x3)  =  U33_ga(x1, x3)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
U35_ga(x1, x2, x3)  =  U35_ga(x1, x3)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
U36_ga(x1, x2, x3)  =  U36_ga(x1, x3)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
U38_ga(x1, x2, x3)  =  U38_ga(x1, x3)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
U39_ga(x1, x2, x3)  =  U39_ga(x1, x3)
U40_ga(x1, x2, x3)  =  U40_ga(x1, x3)
f1_in_aa(x1, x2)  =  f1_in_aa
U31_aa(x1, x2, x3)  =  U31_aa(x3)
f1_out_aa(x1, x2)  =  f1_out_aa(x1)
U32_aa(x1, x2, x3)  =  U32_aa(x1, x3)
U33_aa(x1, x2, x3)  =  U33_aa(x3)
U34_aa(x1, x2, x3)  =  U34_aa(x1, x3)
U35_aa(x1, x2, x3)  =  U35_aa(x3)
U36_aa(x1, x2, x3)  =  U36_aa(x1, x3)
U37_aa(x1, x2, x3)  =  U37_aa(x1, x3)
U38_aa(x1, x2, x3)  =  U38_aa(x1, x3)
U39_aa(x1, x2, x3)  =  U39_aa(x1, x3)
U40_aa(x1, x2, x3)  =  U40_aa(x1, x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)

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

(701) UsableRulesProof (EQUIVALENT transformation)

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

(702) Obligation:

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

U37_AA(T107, T109, add102_out_ga(T107, T148)) → U39_AA(T107, T109, f129_in_ga(T148, T177))
U39_AA(T107, T109, f129_out_ga(T148, T177)) → F1_IN_AA(T177, T109)
F1_IN_AA(T107, T109) → U35_AA(T107, T109, less71_in_a(T107))
U35_AA(T107, T109, less71_out_a(T107)) → U37_AA(T107, T109, add102_in_ga(T107, T148))

The TRS R consists of the following rules:

f129_in_ga(T185, X419) → U23_ga(T185, X419, less59_in_g(T185))
f129_in_ga(T206, X511) → U25_ga(T206, X511, less71_in_g(T206))
less71_in_a(s(T116)) → U21_a(T116, less77_in_a(T116))
less71_in_a(T145) → U22_a(T145, less77_in_a(T145))
add102_in_ga(s(T154), s(X302)) → U13_ga(T154, X302, add102_in_ga(T154, X302))
add102_in_ga(T159, s(X325)) → U14_ga(T159, X325, add110_in_ga(T159, X325))
U23_ga(T185, X419, less59_out_g(T185)) → f129_out_ga(T185, X419)
U23_ga(T185, X419, less59_out_g(T185)) → U24_ga(T185, X419, add136_in_ag(X419, T185))
U25_ga(T206, X511, less71_out_g(T206)) → f129_out_ga(T206, X511)
U25_ga(T206, X511, less71_out_g(T206)) → U26_ga(T206, X511, add102_in_ga(T206, X509))
U25_ga(T206, X511, less71_out_g(T206)) → U27_ga(T206, X511, add102_in_ga(T206, T210))
U21_a(T116, less77_out_a(T116)) → less71_out_a(s(T116))
U22_a(T145, less77_out_a(T145)) → less71_out_a(T145)
U13_ga(T154, X302, add102_out_ga(T154, X302)) → add102_out_ga(s(T154), s(X302))
U14_ga(T159, X325, add110_out_ga(T159, X325)) → add102_out_ga(T159, s(X325))
less59_in_g(s(T93)) → U9_g(T93, less10_in_g(T93))
less59_in_g(s(T97)) → U10_g(T97, less59_in_g(T97))
U24_ga(T185, X419, add136_out_ag(X419, T185)) → f129_out_ga(T185, X419)
less71_in_g(s(T116)) → U21_g(T116, less77_in_g(T116))
less71_in_g(T145) → U22_g(T145, less77_in_g(T145))
U26_ga(T206, X511, add102_out_ga(T206, X509)) → f129_out_ga(T206, X511)
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U28_ga(T206, X511, f129_in_ga(T210, X510))
U27_ga(T206, X511, add102_out_ga(T206, T210)) → U29_ga(T206, X511, f129_in_ga(T210, T213))
less77_in_a(s(T122)) → U11_a(T122, less83_in_a(T122))
less77_in_a(T141) → U12_a(T141, less83_in_a(T141))
add110_in_ga(s(T165), s(X349)) → U15_ga(T165, X349, add110_in_ga(T165, X349))
add110_in_ga(T169, s(X368)) → U16_ga(T169, X368, add117_in_ga(T169, X368))
U9_g(T93, less10_out_g(T93)) → less59_out_g(s(T93))
U10_g(T97, less59_out_g(T97)) → less59_out_g(s(T97))
add136_in_ag(s(X447), s(T193)) → U18_ag(X447, T193, add136_in_ag(X447, T193))
add136_in_ag(X466, s(T197)) → U19_ag(X466, T197, add143_in_ag(X466, T197))
U21_g(T116, less77_out_g(T116)) → less71_out_g(s(T116))
U22_g(T145, less77_out_g(T145)) → less71_out_g(T145)
U28_ga(T206, X511, f129_out_ga(T210, X510)) → f129_out_ga(T206, X511)
U29_ga(T206, X511, f129_out_ga(T210, T213)) → U30_ga(T206, X511, f129_in_aa(T213, X511))
U11_a(T122, less83_out_a(T122)) → less77_out_a(s(T122))
U12_a(T141, less83_out_a(T141)) → less77_out_a(T141)
U15_ga(T165, X349, add110_out_ga(T165, X349)) → add110_out_ga(s(T165), s(X349))
U16_ga(T169, X368, add117_out_ga(T169, X368)) → add110_out_ga(T169, s(X368))
less10_in_g(s(T19)) → U2_g(T19, less16_in_g(T19))
less10_in_g(s(T25)) → U3_g(T25, less10_in_g(T25))
U18_ag(X447, T193, add136_out_ag(X447, T193)) → add136_out_ag(s(X447), s(T193))
U19_ag(X466, T197, add143_out_ag(X466, T197)) → add136_out_ag(X466, s(T197))
less77_in_g(s(T122)) → U11_g(T122, less83_in_g(T122))
less77_in_g(T141) → U12_g(T141, less83_in_g(T141))
U30_ga(T206, X511, f129_out_aa(T213, X511)) → f129_out_ga(T206, X511)
less83_in_a(0) → less83_out_a(0)
add117_in_ga(0, 0) → add117_out_ga(0, 0)
add117_in_ga(s(T174), s(X388)) → U17_ga(T174, X388, add117_in_ga(T174, X388))
U2_g(T19, less16_out_g(T19)) → less10_out_g(s(T19))
U3_g(T25, less10_out_g(T25)) → less10_out_g(s(T25))
add143_in_ag(0, 0) → add143_out_ag(0, 0)
add143_in_ag(s(X489), s(T202)) → U20_ag(X489, T202, add143_in_ag(X489, T202))
U11_g(T122, less83_out_g(T122)) → less77_out_g(s(T122))
U12_g(T141, less83_out_g(T141)) → less77_out_g(T141)
f129_in_aa(T185, X419) → U23_aa(T185, X419, less59_in_a(T185))
f129_in_aa(T206, X511) → U25_aa(T206, X511, less71_in_a(T206))
U17_ga(T174, X388, add117_out_ga(T174, X388)) → add117_out_ga(s(T174), s(X388))
less16_in_g(s(0)) → less16_out_g(s(0))
less16_in_g(s(T22)) → U1_g(T22, less16_in_g(T22))
U20_ag(X489, T202, add143_out_ag(X489, T202)) → add143_out_ag(s(X489), s(T202))
less83_in_g(0) → less83_out_g(0)
U23_aa(T185, X419, less59_out_a(T185)) → f129_out_aa(T185, X419)
U23_aa(T185, X419, less59_out_a(T185)) → U24_aa(T185, X419, add136_in_ag(X419, T185))
U25_aa(T206, X511, less71_out_a(T206)) → f129_out_aa(T206, X511)
U25_aa(T206, X511, less71_out_a(T206)) → U26_aa(T206, X511, add102_in_ga(T206, X509))
U25_aa(T206, X511, less71_out_a(T206)) → U27_aa(T206, X511, add102_in_ga(T206, T210))
U1_g(T22, less16_out_g(T22)) → less16_out_g(s(T22))
less59_in_a(s(T93)) → U9_a(T93, less10_in_a(T93))
less59_in_a(s(T97)) → U10_a(T97, less59_in_a(T97))
U24_aa(T185, X419, add136_out_ag(X419, T185)) → f129_out_aa(T185, X419)
U26_aa(T206, X511, add102_out_ga(T206, X509)) → f129_out_aa(T206, X511)
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U28_aa(T206, X511, f129_in_ga(T210, X510))
U27_aa(T206, X511, add102_out_ga(T206, T210)) → U29_aa(T206, X511, f129_in_ga(T210, T213))
U9_a(T93, less10_out_a(T93)) → less59_out_a(s(T93))
U10_a(T97, less59_out_a(T97)) → less59_out_a(s(T97))
U28_aa(T206, X511, f129_out_ga(T210, X510)) → f129_out_aa(T206, X511)
U29_aa(T206, X511, f129_out_ga(T210, T213)) → U30_aa(T206, X511, f129_in_aa(T213, X511))
less10_in_a(s(T19)) → U2_a(T19, less16_in_a(T19))
less10_in_a(s(T25)) → U3_a(T25, less10_in_a(T25))
U30_aa(T206, X511, f129_out_aa(T213, X511)) → f129_out_aa(T206, X511)
U2_a(T19, less16_out_a(T19)) → less10_out_a(s(T19))
U3_a(T25, less10_out_a(T25)) → less10_out_a(s(T25))
less16_in_a(s(0)) → less16_out_a(s(0))
less16_in_a(s(T22)) → U1_a(T22, less16_in_a(T22))
U1_a(T22, less16_out_a(T22)) → less16_out_a(s(T22))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less10_in_g(x1)  =  less10_in_g(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
less16_in_g(x1)  =  less16_in_g(x1)
0  =  0
less16_out_g(x1)  =  less16_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
less10_out_g(x1)  =  less10_out_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
less59_in_g(x1)  =  less59_in_g(x1)
U9_g(x1, x2)  =  U9_g(x1, x2)
less59_out_g(x1)  =  less59_out_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
less71_in_g(x1)  =  less71_in_g(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
less77_in_g(x1)  =  less77_in_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
less83_in_g(x1)  =  less83_in_g(x1)
less83_out_g(x1)  =  less83_out_g(x1)
less77_out_g(x1)  =  less77_out_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
less71_out_g(x1)  =  less71_out_g(x1)
U22_g(x1, x2)  =  U22_g(x1, x2)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
U13_ga(x1, x2, x3)  =  U13_ga(x1, x3)
U14_ga(x1, x2, x3)  =  U14_ga(x1, x3)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
U16_ga(x1, x2, x3)  =  U16_ga(x1, x3)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
add117_out_ga(x1, x2)  =  add117_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
add110_out_ga(x1, x2)  =  add110_out_ga(x1, x2)
add102_out_ga(x1, x2)  =  add102_out_ga(x1, x2)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
f129_out_ga(x1, x2)  =  f129_out_ga(x1)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
U18_ag(x1, x2, x3)  =  U18_ag(x2, x3)
U19_ag(x1, x2, x3)  =  U19_ag(x2, x3)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
add143_out_ag(x1, x2)  =  add143_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
add136_out_ag(x1, x2)  =  add136_out_ag(x1, x2)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
f129_in_aa(x1, x2)  =  f129_in_aa
U23_aa(x1, x2, x3)  =  U23_aa(x3)
less59_in_a(x1)  =  less59_in_a
U9_a(x1, x2)  =  U9_a(x2)
less10_in_a(x1)  =  less10_in_a
U2_a(x1, x2)  =  U2_a(x2)
less16_in_a(x1)  =  less16_in_a
less16_out_a(x1)  =  less16_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
less10_out_a(x1)  =  less10_out_a(x1)
U3_a(x1, x2)  =  U3_a(x2)
less59_out_a(x1)  =  less59_out_a(x1)
U10_a(x1, x2)  =  U10_a(x2)
f129_out_aa(x1, x2)  =  f129_out_aa(x1)
U24_aa(x1, x2, x3)  =  U24_aa(x1, x3)
U25_aa(x1, x2, x3)  =  U25_aa(x3)
less71_in_a(x1)  =  less71_in_a
U21_a(x1, x2)  =  U21_a(x2)
less77_in_a(x1)  =  less77_in_a
U11_a(x1, x2)  =  U11_a(x2)
less83_in_a(x1)  =  less83_in_a
less83_out_a(x1)  =  less83_out_a(x1)
less77_out_a(x1)  =  less77_out_a(x1)
U12_a(x1, x2)  =  U12_a(x2)
less71_out_a(x1)  =  less71_out_a(x1)
U22_a(x1, x2)  =  U22_a(x2)
U26_aa(x1, x2, x3)  =  U26_aa(x1, x3)
U27_aa(x1, x2, x3)  =  U27_aa(x1, x3)
U28_aa(x1, x2, x3)  =  U28_aa(x1, x3)
U29_aa(x1, x2, x3)  =  U29_aa(x1, x3)
U30_aa(x1, x2, x3)  =  U30_aa(x1, x3)
F1_IN_AA(x1, x2)  =  F1_IN_AA
U35_AA(x1, x2, x3)  =  U35_AA(x3)
U37_AA(x1, x2, x3)  =  U37_AA(x1, x3)
U39_AA(x1, x2, x3)  =  U39_AA(x1, x3)

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