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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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:

Triples:

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(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(s(T127)) :- less93(T127).
less83(T136) :- less93(T136).
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(s(T174), s(X388)) :- add117(T174, X388).
add136(s(X447), s(T193)) :- add136(X447, T193).
add136(X466, s(T197)) :- add143(X466, T197).
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) :- ','(lessc59(T185), add136(X419, T185)).
f129(T206, X511) :- less71(T206).
f129(T206, X511) :- ','(lessc71(T206), add102(T206, X509)).
f129(T206, X511) :- ','(lessc71(T206), ','(addc102(T206, T210), f129(T210, X510))).
f129(T206, X511) :- ','(lessc71(T206), ','(addc102(T206, T210), ','(fc129(T210, T213), f129(T213, X511)))).
f1(s(T12), T7) :- less10(T12).
f1(s(T12), T7) :- ','(lessc10(T12), add11(T7, T12)).
f1(s(T86), T7) :- less59(T86).
f1(s(T86), T7) :- ','(lessc59(T86), add11(T7, T86)).
f1(T107, T109) :- less71(T107).
f1(T107, T109) :- ','(lessc71(T107), add102(T107, X193)).
f1(T107, T109) :- ','(lessc71(T107), ','(addc102(T107, T148), f129(T148, X194))).
f1(T107, T109) :- ','(lessc71(T107), ','(addc102(T107, T148), ','(fc129(T148, T177), f1(T177, T109)))).

Clauses:

lessc16(s(0)).
lessc16(s(T22)) :- lessc16(T22).
lessc10(s(T19)) :- lessc16(T19).
lessc10(s(T25)) :- lessc10(T25).
addc33(s(T52), s(T51)) :- addc33(T52, T51).
addc33(T61, s(T60)) :- addc41(T61, T60).
addc41(0, 0).
addc41(s(T72), s(T71)) :- addc41(T72, T71).
addc11(s(T39), T38) :- addc33(T39, T38).
addc11(T81, T80) :- addc41(T81, T80).
lessc59(s(T93)) :- lessc10(T93).
lessc59(s(T97)) :- lessc59(T97).
lessc83(0).
lessc83(s(T127)) :- lessc93(T127).
lessc83(T136) :- lessc93(T136).
lessc77(s(T122)) :- lessc83(T122).
lessc77(T141) :- lessc83(T141).
addc102(s(T154), s(X302)) :- addc102(T154, X302).
addc102(T159, s(X325)) :- addc110(T159, X325).
addc110(s(T165), s(X349)) :- addc110(T165, X349).
addc110(T169, s(X368)) :- addc117(T169, X368).
addc117(0, 0).
addc117(s(T174), s(X388)) :- addc117(T174, X388).
fc1(s(T12), T7) :- ','(lessc10(T12), addc11(T7, T12)).
fc1(s(T86), T7) :- ','(lessc59(T86), addc11(T7, T86)).
fc1(T107, T109) :- ','(lessc71(T107), ','(addc102(T107, T148), ','(fc129(T148, T177), fc1(T177, T109)))).
addc136(s(X447), s(T193)) :- addc136(X447, T193).
addc136(X466, s(T197)) :- addc143(X466, T197).
addc143(0, 0).
addc143(s(X489), s(T202)) :- addc143(X489, T202).
lessc71(s(T116)) :- lessc77(T116).
lessc71(T145) :- lessc77(T145).
fc129(T185, X419) :- ','(lessc59(T185), addc136(X419, T185)).
fc129(T206, X511) :- ','(lessc71(T206), ','(addc102(T206, T210), ','(fc129(T210, T213), fc129(T213, X511)))).

Afs:

f1(x1, x2)  =  f1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

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(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).
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(s(T174), s(X388)) :- add117(T174, X388).
add136(s(X447), s(T193)) :- add136(X447, T193).
add136(X466, s(T197)) :- add143(X466, T197).
add143(s(X489), s(T202)) :- add143(X489, T202).
f129(T185, X419) :- less59(T185).
f129(T185, X419) :- ','(lessc59(T185), add136(X419, T185)).
f129(T206, X511) :- ','(lessc71(T206), add102(T206, X509)).
f129(T206, X511) :- ','(lessc71(T206), ','(addc102(T206, T210), f129(T210, X510))).
f129(T206, X511) :- ','(lessc71(T206), ','(addc102(T206, T210), ','(fc129(T210, T213), f129(T213, X511)))).
f1(s(T12), T7) :- less10(T12).
f1(s(T12), T7) :- ','(lessc10(T12), add11(T7, T12)).
f1(s(T86), T7) :- less59(T86).
f1(s(T86), T7) :- ','(lessc59(T86), add11(T7, T86)).
f1(T107, T109) :- ','(lessc71(T107), add102(T107, X193)).
f1(T107, T109) :- ','(lessc71(T107), ','(addc102(T107, T148), f129(T148, X194))).
f1(T107, T109) :- ','(lessc71(T107), ','(addc102(T107, T148), ','(fc129(T148, T177), f1(T177, T109)))).

Clauses:

lessc16(s(0)).
lessc16(s(T22)) :- lessc16(T22).
lessc10(s(T19)) :- lessc16(T19).
lessc10(s(T25)) :- lessc10(T25).
addc33(s(T52), s(T51)) :- addc33(T52, T51).
addc33(T61, s(T60)) :- addc41(T61, T60).
addc41(0, 0).
addc41(s(T72), s(T71)) :- addc41(T72, T71).
addc11(s(T39), T38) :- addc33(T39, T38).
addc11(T81, T80) :- addc41(T81, T80).
lessc59(s(T93)) :- lessc10(T93).
lessc59(s(T97)) :- lessc59(T97).
lessc83(0).
lessc77(s(T122)) :- lessc83(T122).
lessc77(T141) :- lessc83(T141).
addc102(s(T154), s(X302)) :- addc102(T154, X302).
addc102(T159, s(X325)) :- addc110(T159, X325).
addc110(s(T165), s(X349)) :- addc110(T165, X349).
addc110(T169, s(X368)) :- addc117(T169, X368).
addc117(0, 0).
addc117(s(T174), s(X388)) :- addc117(T174, X388).
fc1(s(T12), T7) :- ','(lessc10(T12), addc11(T7, T12)).
fc1(s(T86), T7) :- ','(lessc59(T86), addc11(T7, T86)).
fc1(T107, T109) :- ','(lessc71(T107), ','(addc102(T107, T148), ','(fc129(T148, T177), fc1(T177, T109)))).
addc136(s(X447), s(T193)) :- addc136(X447, T193).
addc136(X466, s(T197)) :- addc143(X466, T197).
addc143(0, 0).
addc143(s(X489), s(T202)) :- addc143(X489, T202).
lessc71(s(T116)) :- lessc77(T116).
lessc71(T145) :- lessc77(T145).
fc129(T185, X419) :- ','(lessc59(T185), addc136(X419, T185)).
fc129(T206, X511) :- ','(lessc71(T206), ','(addc102(T206, T210), ','(fc129(T210, T213), fc129(T213, X511)))).

Afs:

f1(x1, x2)  =  f1(x1)

(5) TriplesToPiDPProof (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)
less10_in: (b)
less16_in: (b)
lessc10_in: (b)
lessc16_in: (b)
add11_in: (f,b)
add33_in: (f,b)
add41_in: (f,b)
less59_in: (b)
lessc59_in: (b)
lessc71_in: (b)
lessc77_in: (b)
add102_in: (b,f)
add110_in: (b,f)
add117_in: (b,f)
addc102_in: (b,f)
addc110_in: (b,f)
addc117_in: (b,f)
f129_in: (b,f)
add136_in: (f,b)
add143_in: (f,b)
fc129_in: (b,f)
addc136_in: (f,b)
addc143_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

F1_IN_GA(s(T12), T7) → U28_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)
F1_IN_GA(s(T12), T7) → U29_GA(T12, T7, lessc10_in_g(T12))
U29_GA(T12, T7, lessc10_out_g(T12)) → U30_GA(T12, T7, add11_in_ag(T7, T12))
U29_GA(T12, T7, lessc10_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) → U31_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)
F1_IN_GA(s(T86), T7) → U32_GA(T86, T7, lessc59_in_g(T86))
U32_GA(T86, T7, lessc59_out_g(T86)) → U33_GA(T86, T7, add11_in_ag(T7, T86))
U32_GA(T86, T7, lessc59_out_g(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_GA(T107, T109) → U34_GA(T107, T109, lessc71_in_g(T107))
U34_GA(T107, T109, lessc71_out_g(T107)) → U35_GA(T107, T109, add102_in_ga(T107, X193))
U34_GA(T107, T109, lessc71_out_g(T107)) → ADD102_IN_GA(T107, X193)
ADD102_IN_GA(s(T154), s(X302)) → U11_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)) → U12_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)) → U13_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)) → U14_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)) → U15_GA(T174, X388, add117_in_ga(T174, X388))
ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)
U34_GA(T107, T109, lessc71_out_g(T107)) → U36_GA(T107, T109, addc102_in_ga(T107, T148))
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → U37_GA(T107, T109, f129_in_ga(T148, X194))
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
F129_IN_GA(T185, X419) → U19_GA(T185, X419, less59_in_g(T185))
F129_IN_GA(T185, X419) → LESS59_IN_G(T185)
F129_IN_GA(T185, X419) → U20_GA(T185, X419, lessc59_in_g(T185))
U20_GA(T185, X419, lessc59_out_g(T185)) → U21_GA(T185, X419, add136_in_ag(X419, T185))
U20_GA(T185, X419, lessc59_out_g(T185)) → ADD136_IN_AG(X419, T185)
ADD136_IN_AG(s(X447), s(T193)) → U16_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)) → U17_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)) → U18_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) → U22_GA(T206, X511, lessc71_in_g(T206))
U22_GA(T206, X511, lessc71_out_g(T206)) → U23_GA(T206, X511, add102_in_ga(T206, X509))
U22_GA(T206, X511, lessc71_out_g(T206)) → ADD102_IN_GA(T206, X509)
U22_GA(T206, X511, lessc71_out_g(T206)) → U24_GA(T206, X511, addc102_in_ga(T206, T210))
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → U25_GA(T206, X511, f129_in_ga(T210, X510))
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → U26_GA(T206, X511, fc129_in_ga(T210, T213))
U26_GA(T206, X511, fc129_out_ga(T210, T213)) → U27_GA(T206, X511, f129_in_ga(T213, X511))
U26_GA(T206, X511, fc129_out_ga(T210, T213)) → F129_IN_GA(T213, X511)
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → U38_GA(T107, T109, fc129_in_ga(T148, T177))
U38_GA(T107, T109, fc129_out_ga(T148, T177)) → U39_GA(T107, T109, f1_in_ga(T177, T109))
U38_GA(T107, T109, fc129_out_ga(T148, T177)) → F1_IN_GA(T177, T109)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
less10_in_g(x1)  =  less10_in_g(x1)
less16_in_g(x1)  =  less16_in_g(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
less59_in_g(x1)  =  less59_in_g(x1)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U28_GA(x1, x2, x3)  =  U28_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)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_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)
U31_GA(x1, x2, x3)  =  U31_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)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
U37_GA(x1, x2, x3)  =  U37_GA(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)
U16_AG(x1, x2, x3)  =  U16_AG(x2, x3)
U17_AG(x1, x2, x3)  =  U17_AG(x2, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)
U18_AG(x1, x2, x3)  =  U18_AG(x2, x3)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, 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)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x1, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

F1_IN_GA(s(T12), T7) → U28_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)
F1_IN_GA(s(T12), T7) → U29_GA(T12, T7, lessc10_in_g(T12))
U29_GA(T12, T7, lessc10_out_g(T12)) → U30_GA(T12, T7, add11_in_ag(T7, T12))
U29_GA(T12, T7, lessc10_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) → U31_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)
F1_IN_GA(s(T86), T7) → U32_GA(T86, T7, lessc59_in_g(T86))
U32_GA(T86, T7, lessc59_out_g(T86)) → U33_GA(T86, T7, add11_in_ag(T7, T86))
U32_GA(T86, T7, lessc59_out_g(T86)) → ADD11_IN_AG(T7, T86)
F1_IN_GA(T107, T109) → U34_GA(T107, T109, lessc71_in_g(T107))
U34_GA(T107, T109, lessc71_out_g(T107)) → U35_GA(T107, T109, add102_in_ga(T107, X193))
U34_GA(T107, T109, lessc71_out_g(T107)) → ADD102_IN_GA(T107, X193)
ADD102_IN_GA(s(T154), s(X302)) → U11_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)) → U12_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)) → U13_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)) → U14_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)) → U15_GA(T174, X388, add117_in_ga(T174, X388))
ADD117_IN_GA(s(T174), s(X388)) → ADD117_IN_GA(T174, X388)
U34_GA(T107, T109, lessc71_out_g(T107)) → U36_GA(T107, T109, addc102_in_ga(T107, T148))
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → U37_GA(T107, T109, f129_in_ga(T148, X194))
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → F129_IN_GA(T148, X194)
F129_IN_GA(T185, X419) → U19_GA(T185, X419, less59_in_g(T185))
F129_IN_GA(T185, X419) → LESS59_IN_G(T185)
F129_IN_GA(T185, X419) → U20_GA(T185, X419, lessc59_in_g(T185))
U20_GA(T185, X419, lessc59_out_g(T185)) → U21_GA(T185, X419, add136_in_ag(X419, T185))
U20_GA(T185, X419, lessc59_out_g(T185)) → ADD136_IN_AG(X419, T185)
ADD136_IN_AG(s(X447), s(T193)) → U16_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)) → U17_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)) → U18_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) → U22_GA(T206, X511, lessc71_in_g(T206))
U22_GA(T206, X511, lessc71_out_g(T206)) → U23_GA(T206, X511, add102_in_ga(T206, X509))
U22_GA(T206, X511, lessc71_out_g(T206)) → ADD102_IN_GA(T206, X509)
U22_GA(T206, X511, lessc71_out_g(T206)) → U24_GA(T206, X511, addc102_in_ga(T206, T210))
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → U25_GA(T206, X511, f129_in_ga(T210, X510))
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → U26_GA(T206, X511, fc129_in_ga(T210, T213))
U26_GA(T206, X511, fc129_out_ga(T210, T213)) → U27_GA(T206, X511, f129_in_ga(T213, X511))
U26_GA(T206, X511, fc129_out_ga(T210, T213)) → F129_IN_GA(T213, X511)
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → U38_GA(T107, T109, fc129_in_ga(T148, T177))
U38_GA(T107, T109, fc129_out_ga(T148, T177)) → U39_GA(T107, T109, f1_in_ga(T177, T109))
U38_GA(T107, T109, fc129_out_ga(T148, T177)) → F1_IN_GA(T177, T109)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
f1_in_ga(x1, x2)  =  f1_in_ga(x1)
s(x1)  =  s(x1)
less10_in_g(x1)  =  less10_in_g(x1)
less16_in_g(x1)  =  less16_in_g(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
add11_in_ag(x1, x2)  =  add11_in_ag(x2)
add33_in_ag(x1, x2)  =  add33_in_ag(x2)
add41_in_ag(x1, x2)  =  add41_in_ag(x2)
less59_in_g(x1)  =  less59_in_g(x1)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
add102_in_ga(x1, x2)  =  add102_in_ga(x1)
add110_in_ga(x1, x2)  =  add110_in_ga(x1)
add117_in_ga(x1, x2)  =  add117_in_ga(x1)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
f129_in_ga(x1, x2)  =  f129_in_ga(x1)
add136_in_ag(x1, x2)  =  add136_in_ag(x2)
add143_in_ag(x1, x2)  =  add143_in_ag(x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U28_GA(x1, x2, x3)  =  U28_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)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_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)
U31_GA(x1, x2, x3)  =  U31_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)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
U37_GA(x1, x2, x3)  =  U37_GA(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)
U16_AG(x1, x2, x3)  =  U16_AG(x2, x3)
U17_AG(x1, x2, x3)  =  U17_AG(x2, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)
U18_AG(x1, x2, x3)  =  U18_AG(x2, x3)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, 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)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x1, x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 12 SCCs with 50 less nodes.

(8) Complex Obligation (AND)

(9) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD143_IN_AG(x1, x2)  =  ADD143_IN_AG(x2)

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:

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

(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:

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

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(15) YES

(16) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD136_IN_AG(x1, x2)  =  ADD136_IN_AG(x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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

(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:

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

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(22) YES

(23) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD117_IN_GA(x1, x2)  =  ADD117_IN_GA(x1)

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:

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

(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:

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

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(29) YES

(30) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD110_IN_GA(x1, x2)  =  ADD110_IN_GA(x1)

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:

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

(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:

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

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:

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

(36) YES

(37) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD102_IN_GA(x1, x2)  =  ADD102_IN_GA(x1)

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:

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

(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:

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

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:

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

(43) YES

(44) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD41_IN_AG(x1, x2)  =  ADD41_IN_AG(x2)

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:

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

(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:

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

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:

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

(50) YES

(51) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
ADD33_IN_AG(x1, x2)  =  ADD33_IN_AG(x2)

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:

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

(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:

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

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:

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

(57) YES

(58) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
LESS16_IN_G(x1)  =  LESS16_IN_G(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:

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

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

(61) PiDPToQDPProof (EQUIVALENT 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:

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

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:

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

(64) YES

(65) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
LESS10_IN_G(x1)  =  LESS10_IN_G(x1)

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

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

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

(68) PiDPToQDPProof (EQUIVALENT 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:

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

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:

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

(71) YES

(72) 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:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
LESS59_IN_G(x1)  =  LESS59_IN_G(x1)

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:

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

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

(75) PiDPToQDPProof (EQUIVALENT 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:

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

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:

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

(78) YES

(79) Obligation:

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

F129_IN_GA(T206, X511) → U22_GA(T206, X511, lessc71_in_g(T206))
U22_GA(T206, X511, lessc71_out_g(T206)) → U24_GA(T206, X511, addc102_in_ga(T206, T210))
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → F129_IN_GA(T210, X510)
U24_GA(T206, X511, addc102_out_ga(T206, T210)) → U26_GA(T206, X511, fc129_in_ga(T210, T213))
U26_GA(T206, X511, fc129_out_ga(T210, T213)) → F129_IN_GA(T213, X511)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
F129_IN_GA(x1, x2)  =  F129_IN_GA(x1)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, x3)
U26_GA(x1, x2, x3)  =  U26_GA(x1, x3)

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

(80) PiDPToQDPProof (SOUND transformation)

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

(81) Obligation:

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

F129_IN_GA(T206) → U22_GA(T206, lessc71_in_g(T206))
U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(82) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(T206) → U22_GA(T206, lessc71_in_g(T206)) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, lessc77_in_g(x0)))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, lessc77_in_g(x0)))

(83) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, lessc77_in_g(x0)))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, lessc77_in_g(x0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(84) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, lessc77_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(x0))) → U22_GA(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))

(85) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, lessc77_in_g(x0)))
F129_IN_GA(s(s(x0))) → U22_GA(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(86) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, lessc77_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))

(87) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(x0))) → U22_GA(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(88) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(s(x0))) → U22_GA(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0))))

(89) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(90) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), U69_g(s(0), lessc77_out_g(s(0))))

(91) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), U69_g(s(0), lessc77_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(92) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), U69_g(s(0), lessc77_out_g(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))

(93) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(94) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U22_GA(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U22_GA(s(0), U69_g(0, U52_g(0, lessc83_out_g(0))))

(95) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), U69_g(0, U52_g(0, lessc83_out_g(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(96) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U22_GA(s(0), U69_g(0, U52_g(0, lessc83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U22_GA(s(0), U69_g(0, lessc77_out_g(0)))

(97) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), U69_g(0, lessc77_out_g(0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(98) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U22_GA(s(0), U69_g(0, lessc77_out_g(0))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))

(99) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(100) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(s(x0)) → U22_GA(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U22_GA(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0))))

(101) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(s(0)) → U22_GA(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(102) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U22_GA(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U22_GA(s(0), U70_g(s(0), lessc77_out_g(s(0))))

(103) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(s(0)) → U22_GA(s(0), U70_g(s(0), lessc77_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(104) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(s(0)) → U22_GA(s(0), U70_g(s(0), lessc77_out_g(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))

(105) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(106) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F129_IN_GA(x0) → U22_GA(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U22_GA(0, U70_g(0, U52_g(0, lessc83_out_g(0))))

(107) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, U70_g(0, U52_g(0, lessc83_out_g(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(108) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(0) → U22_GA(0, U70_g(0, U52_g(0, lessc83_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U22_GA(0, U70_g(0, lessc77_out_g(0)))

(109) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, U70_g(0, lessc77_out_g(0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(110) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F129_IN_GA(0) → U22_GA(0, U70_g(0, lessc77_out_g(0))) at position [1] we obtained the following new rules [LPAR04]:

F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))

(111) Obligation:

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

U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206))
U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(112) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U22_GA(T206, lessc71_out_g(T206)) → U24_GA(T206, addc102_in_ga(T206)) we obtained the following new rules [LPAR04]:

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U22_GA(0, lessc71_out_g(0)) → U24_GA(0, addc102_in_ga(0))

(113) Obligation:

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

U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210)
U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U22_GA(0, lessc71_out_g(0)) → U24_GA(0, addc102_in_ga(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(114) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U24_GA(T206, addc102_out_ga(T206, T210)) → F129_IN_GA(T210) we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(0, addc102_out_ga(0, x1)) → F129_IN_GA(x1)

(115) Obligation:

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

U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210))
U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U22_GA(0, lessc71_out_g(0)) → U24_GA(0, addc102_in_ga(0))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(0, addc102_out_ga(0, x1)) → F129_IN_GA(x1)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(116) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U24_GA(T206, addc102_out_ga(T206, T210)) → U26_GA(T206, fc129_in_ga(T210)) we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U24_GA(0, addc102_out_ga(0, x1)) → U26_GA(0, fc129_in_ga(x1))

(117) Obligation:

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

U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U22_GA(0, lessc71_out_g(0)) → U24_GA(0, addc102_in_ga(0))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(0, addc102_out_ga(0, x1)) → F129_IN_GA(x1)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U24_GA(0, addc102_out_ga(0, x1)) → U26_GA(0, fc129_in_ga(x1))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(118) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U26_GA(T206, fc129_out_ga(T210, T213)) → F129_IN_GA(T213) we obtained the following new rules [LPAR04]:

U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(0, fc129_out_ga(x1, x2)) → F129_IN_GA(x2)

(119) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U22_GA(0, lessc71_out_g(0)) → U24_GA(0, addc102_in_ga(0))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(0, addc102_out_ga(0, x1)) → F129_IN_GA(x1)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U24_GA(0, addc102_out_ga(0, x1)) → U26_GA(0, fc129_in_ga(x1))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(0, fc129_out_ga(x1, x2)) → F129_IN_GA(x2)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22_GA(0, lessc71_out_g(0)) → U24_GA(0, addc102_in_ga(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 0 \ \ 1 0 /

·

x1

POL(0) =

/ 1 \ \ 0 /

POL(U22_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(lessc71_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U24_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(addc102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(addc102_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(U26_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(fc129_in_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(fc129_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U53_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U54_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(addc110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U71_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(lessc59_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U73_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U69_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(lessc77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U70_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 1 0 \ \ 1 1 /

·

x2

POL(U74_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U75_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U76_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U49_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(lessc10_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U50_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

POL(lessc59_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U72_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(addc136_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U42_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc16_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U43_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc10_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(lessc16_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U41_g(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(U51_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(lessc83_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U52_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(lessc77_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(lessc83_out_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U55_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U56_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(addc117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(addc110_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(addc117_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 1 1 /

·

x2

POL(U57_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U66_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U67_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(addc143_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(addc136_out_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(addc143_out_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U68_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

The following usable rules [FROCOS05] were oriented:

addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))

(121) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
F129_IN_GA(0) → U22_GA(0, lessc71_out_g(0))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(0, addc102_out_ga(0, x1)) → F129_IN_GA(x1)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U24_GA(0, addc102_out_ga(0, x1)) → U26_GA(0, fc129_in_ga(x1))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(0, fc129_out_ga(x1, x2)) → F129_IN_GA(x2)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(122) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(123) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(124) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), addc102_in_ga(s(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))

(125) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(126) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), addc102_in_ga(s(0))) at position [1] we obtained the following new rules [LPAR04]:

U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))

(127) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(128) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), x1)) → U26_GA(s(0), fc129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, lessc71_in_g(x0)))

(129) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, lessc71_in_g(x0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(130) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → U26_GA(s(s(0)), fc129_in_ga(x1)) at position [1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, lessc71_in_g(x0)))

(131) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, lessc71_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, lessc71_in_g(x0)))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(132) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, lessc71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))

(133) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, lessc71_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(134) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, lessc71_in_g(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))

(135) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(136) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(s(x0)))) → U26_GA(s(0), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))

(137) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(x0)))) → U26_GA(s(0), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(138) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, lessc77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))

(139) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(x0)))) → U26_GA(s(0), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(140) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, lessc77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))

(141) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, lessc77_in_g(x0))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(x0)))) → U26_GA(s(0), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(142) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, lessc77_in_g(x0)))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))

(143) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(s(x0)))) → U26_GA(s(0), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(144) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(s(x0)))) → U26_GA(s(0), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0)))))

(145) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(146) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), U69_g(s(0), lessc77_out_g(s(0)))))

(147) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), U69_g(s(0), lessc77_out_g(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(148) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), U69_g(s(0), lessc77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))

(149) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(150) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U69_g(0, U52_g(0, lessc83_out_g(0)))))

(151) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U69_g(0, U52_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(152) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U69_g(0, U52_g(0, lessc83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U69_g(0, lessc77_out_g(0))))

(153) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U69_g(0, lessc77_out_g(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(154) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U69_g(0, lessc77_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))

(155) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(156) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(x0))) → U26_GA(s(0), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0)))))

(157) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(158) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U70_g(s(0), lessc77_out_g(s(0)))))

(159) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U70_g(s(0), lessc77_out_g(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(160) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), U70_g(s(0), lessc77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))

(161) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(162) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), 0)) → U26_GA(s(0), U73_ga(0, U70_g(0, U52_g(0, lessc83_out_g(0)))))

(163) Obligation:

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

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(0), addc102_out_ga(s(0), 0)) → U26_GA(s(0), U73_ga(0, U70_g(0, U52_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(164) DependencyGraphProof (EQUIVALENT transformation)

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

(165) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(166) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(x0)))) → U26_GA(s(s(0)), U73_ga(s(s(x0)), U69_g(s(x0), U51_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0)))))

(167) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(168) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), U69_g(s(0), U51_g(0, lessc83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), U69_g(s(0), lessc77_out_g(s(0)))))

(169) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), U69_g(s(0), lessc77_out_g(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(170) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), U69_g(s(0), lessc77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))

(171) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(172) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U69_g(x0, U52_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U69_g(0, U52_g(0, lessc83_out_g(0)))))

(173) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U69_g(0, U52_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(174) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U69_g(0, U52_g(0, lessc83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U69_g(0, lessc77_out_g(0))))

(175) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U69_g(0, lessc77_out_g(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(176) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U69_g(0, lessc77_out_g(0)))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))

(177) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(178) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(x0))) → U26_GA(s(s(0)), U73_ga(s(x0), U70_g(s(x0), U51_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0)))))

(179) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(180) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U70_g(s(0), U51_g(0, lessc83_out_g(0))))) at position [1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U70_g(s(0), lessc77_out_g(s(0)))))

(181) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U70_g(s(0), lessc77_out_g(s(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(182) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), U70_g(s(0), lessc77_out_g(s(0))))) at position [1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))

(183) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0)))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(184) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U73_ga(x0, U70_g(x0, U52_g(x0, lessc83_in_g(x0))))) at position [1,1,1,1] we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), 0)) → U26_GA(s(s(0)), U73_ga(0, U70_g(0, U52_g(0, lessc83_out_g(0)))))

(185) Obligation:

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

F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), 0)) → U26_GA(s(s(0)), U73_ga(0, U70_g(0, U52_g(0, lessc83_out_g(0)))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(186) DependencyGraphProof (EQUIVALENT transformation)

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

(187) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1)
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(188) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))

(189) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(190) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U26_GA(s(s(0)), fc129_out_ga(x1, x2)) → F129_IN_GA(x2) we obtained the following new rules [LPAR04]:

U26_GA(s(s(0)), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(s(0)), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))

(191) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1)
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U26_GA(s(s(0)), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(s(0)), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(192) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U24_GA(s(0), addc102_out_ga(s(0), x1)) → F129_IN_GA(x1) we obtained the following new rules [LPAR04]:

U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

(193) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2)
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U26_GA(s(s(0)), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(s(0)), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(194) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U26_GA(s(0), fc129_out_ga(x1, x2)) → F129_IN_GA(x2) we obtained the following new rules [LPAR04]:

U26_GA(s(0), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(0), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))

(195) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U26_GA(s(s(0)), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(s(0)), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U24_GA(s(0), addc102_out_ga(s(0), s(0))) → F129_IN_GA(s(0))
U26_GA(s(0), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(0), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(196) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U24_GA(s(0), addc102_out_ga(s(0), s(0))) → U26_GA(s(0), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → U26_GA(s(s(0)), U73_ga(s(0), lessc71_out_g(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(0))) → F129_IN_GA(s(0))
U24_GA(s(0), addc102_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(U22_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(0) =

/ 1 \ \ 0 /

POL(lessc71_out_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U24_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U53_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(addc102_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(F129_IN_GA(x1)) =

0

+

[ 0, 0 ]

·

x1

POL(U54_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(addc110_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(addc102_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(U26_GA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U71_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 1 0 /

·

x2

POL(lessc59_in_g(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(U73_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(fc129_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U55_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U56_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(addc117_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(addc110_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(U49_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(lessc10_in_g(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U50_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc59_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL(U72_ga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(addc136_in_ag(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U74_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(fc129_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(lessc71_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U69_g(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc77_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U70_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(U75_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U76_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(U42_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc16_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U43_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(lessc10_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(lessc16_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U41_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U51_g(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(lessc83_in_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U52_g(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(lessc77_out_g(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(lessc83_out_g(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(addc117_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U57_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U66_ag(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U67_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(addc143_in_ag(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(addc136_out_ag(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(addc143_out_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U68_ag(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

The following usable rules [FROCOS05] were oriented:

addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))

(197) Obligation:

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

U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U53_ga(s(0), addc102_in_ga(s(0))))
F129_IN_GA(s(s(0))) → U22_GA(s(s(0)), lessc71_out_g(s(s(0))))
U22_GA(s(s(0)), lessc71_out_g(s(s(0)))) → U24_GA(s(s(0)), U54_ga(s(s(0)), addc110_in_ga(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), x0)) → U26_GA(s(s(0)), U71_ga(x0, lessc59_in_g(x0)))
F129_IN_GA(s(0)) → U22_GA(s(0), lessc71_out_g(s(0)))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U53_ga(0, addc102_in_ga(0)))
U24_GA(s(0), addc102_out_ga(s(0), x0)) → U26_GA(s(0), U71_ga(x0, lessc59_in_g(x0)))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → U26_GA(s(0), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U22_GA(s(0), lessc71_out_g(s(0))) → U24_GA(s(0), U54_ga(s(0), addc110_in_ga(s(0))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → U26_GA(s(s(0)), U73_ga(s(s(0)), lessc71_out_g(s(s(0)))))
U24_GA(s(s(0)), addc102_out_ga(s(s(0)), s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(s(0)), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(s(0)), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))
U24_GA(s(0), addc102_out_ga(s(0), s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(0), fc129_out_ga(x0, s(s(0)))) → F129_IN_GA(s(s(0)))
U26_GA(s(0), fc129_out_ga(x0, s(0))) → F129_IN_GA(s(0))

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154)) → U53_ga(T154, addc102_in_ga(T154))
addc102_in_ga(T159) → U54_ga(T159, addc110_in_ga(T159))
addc110_in_ga(s(T165)) → U55_ga(T165, addc110_in_ga(T165))
addc110_in_ga(T169) → U56_ga(T169, addc117_in_ga(T169))
addc117_in_ga(0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174)) → U57_ga(T174, addc117_in_ga(T174))
U57_ga(T174, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185) → U71_ga(T185, lessc59_in_g(T185))
U71_ga(T185, lessc59_out_g(T185)) → U72_ga(T185, addc136_in_ag(T185))
addc136_in_ag(s(T193)) → U66_ag(T193, addc136_in_ag(T193))
addc136_in_ag(s(T197)) → U67_ag(T197, addc143_in_ag(T197))
addc143_in_ag(0) → addc143_out_ag(0, 0)
addc143_in_ag(s(T202)) → U68_ag(T202, addc143_in_ag(T202))
U68_ag(T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206) → U73_ga(T206, lessc71_in_g(T206))
U73_ga(T206, lessc71_out_g(T206)) → U74_ga(T206, addc102_in_ga(T206))
U74_ga(T206, addc102_out_ga(T206, T210)) → U75_ga(T206, fc129_in_ga(T210))
U75_ga(T206, fc129_out_ga(T210, T213)) → U76_ga(T206, fc129_in_ga(T213))
U76_ga(T206, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The set Q consists of the following terms:

lessc10_in_g(x0)
lessc16_in_g(x0)
U41_g(x0, x1)
U42_g(x0, x1)
U43_g(x0, x1)
lessc59_in_g(x0)
U49_g(x0, x1)
U50_g(x0, x1)
lessc71_in_g(x0)
lessc77_in_g(x0)
lessc83_in_g(x0)
U51_g(x0, x1)
U52_g(x0, x1)
U69_g(x0, x1)
U70_g(x0, x1)
addc102_in_ga(x0)
addc110_in_ga(x0)
addc117_in_ga(x0)
U57_ga(x0, x1)
U56_ga(x0, x1)
U55_ga(x0, x1)
U54_ga(x0, x1)
U53_ga(x0, x1)
fc129_in_ga(x0)
U71_ga(x0, x1)
addc136_in_ag(x0)
addc143_in_ag(x0)
U68_ag(x0, x1)
U67_ag(x0, x1)
U66_ag(x0, x1)
U72_ga(x0, x1)
U73_ga(x0, x1)
U74_ga(x0, x1)
U75_ga(x0, x1)
U76_ga(x0, x1)

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

(198) Obligation:

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

F1_IN_GA(T107, T109) → U34_GA(T107, T109, lessc71_in_g(T107))
U34_GA(T107, T109, lessc71_out_g(T107)) → U36_GA(T107, T109, addc102_in_ga(T107, T148))
U36_GA(T107, T109, addc102_out_ga(T107, T148)) → U38_GA(T107, T109, fc129_in_ga(T148, T177))
U38_GA(T107, T109, fc129_out_ga(T148, T177)) → F1_IN_GA(T177, T109)

The TRS R consists of the following rules:

lessc10_in_g(s(T19)) → U42_g(T19, lessc16_in_g(T19))
lessc16_in_g(s(0)) → lessc16_out_g(s(0))
lessc16_in_g(s(T22)) → U41_g(T22, lessc16_in_g(T22))
U41_g(T22, lessc16_out_g(T22)) → lessc16_out_g(s(T22))
U42_g(T19, lessc16_out_g(T19)) → lessc10_out_g(s(T19))
lessc10_in_g(s(T25)) → U43_g(T25, lessc10_in_g(T25))
U43_g(T25, lessc10_out_g(T25)) → lessc10_out_g(s(T25))
lessc59_in_g(s(T93)) → U49_g(T93, lessc10_in_g(T93))
U49_g(T93, lessc10_out_g(T93)) → lessc59_out_g(s(T93))
lessc59_in_g(s(T97)) → U50_g(T97, lessc59_in_g(T97))
U50_g(T97, lessc59_out_g(T97)) → lessc59_out_g(s(T97))
lessc71_in_g(s(T116)) → U69_g(T116, lessc77_in_g(T116))
lessc77_in_g(s(T122)) → U51_g(T122, lessc83_in_g(T122))
lessc83_in_g(0) → lessc83_out_g(0)
U51_g(T122, lessc83_out_g(T122)) → lessc77_out_g(s(T122))
lessc77_in_g(T141) → U52_g(T141, lessc83_in_g(T141))
U52_g(T141, lessc83_out_g(T141)) → lessc77_out_g(T141)
U69_g(T116, lessc77_out_g(T116)) → lessc71_out_g(s(T116))
lessc71_in_g(T145) → U70_g(T145, lessc77_in_g(T145))
U70_g(T145, lessc77_out_g(T145)) → lessc71_out_g(T145)
addc102_in_ga(s(T154), s(X302)) → U53_ga(T154, X302, addc102_in_ga(T154, X302))
addc102_in_ga(T159, s(X325)) → U54_ga(T159, X325, addc110_in_ga(T159, X325))
addc110_in_ga(s(T165), s(X349)) → U55_ga(T165, X349, addc110_in_ga(T165, X349))
addc110_in_ga(T169, s(X368)) → U56_ga(T169, X368, addc117_in_ga(T169, X368))
addc117_in_ga(0, 0) → addc117_out_ga(0, 0)
addc117_in_ga(s(T174), s(X388)) → U57_ga(T174, X388, addc117_in_ga(T174, X388))
U57_ga(T174, X388, addc117_out_ga(T174, X388)) → addc117_out_ga(s(T174), s(X388))
U56_ga(T169, X368, addc117_out_ga(T169, X368)) → addc110_out_ga(T169, s(X368))
U55_ga(T165, X349, addc110_out_ga(T165, X349)) → addc110_out_ga(s(T165), s(X349))
U54_ga(T159, X325, addc110_out_ga(T159, X325)) → addc102_out_ga(T159, s(X325))
U53_ga(T154, X302, addc102_out_ga(T154, X302)) → addc102_out_ga(s(T154), s(X302))
fc129_in_ga(T185, X419) → U71_ga(T185, X419, lessc59_in_g(T185))
U71_ga(T185, X419, lessc59_out_g(T185)) → U72_ga(T185, X419, addc136_in_ag(X419, T185))
addc136_in_ag(s(X447), s(T193)) → U66_ag(X447, T193, addc136_in_ag(X447, T193))
addc136_in_ag(X466, s(T197)) → U67_ag(X466, T197, addc143_in_ag(X466, T197))
addc143_in_ag(0, 0) → addc143_out_ag(0, 0)
addc143_in_ag(s(X489), s(T202)) → U68_ag(X489, T202, addc143_in_ag(X489, T202))
U68_ag(X489, T202, addc143_out_ag(X489, T202)) → addc143_out_ag(s(X489), s(T202))
U67_ag(X466, T197, addc143_out_ag(X466, T197)) → addc136_out_ag(X466, s(T197))
U66_ag(X447, T193, addc136_out_ag(X447, T193)) → addc136_out_ag(s(X447), s(T193))
U72_ga(T185, X419, addc136_out_ag(X419, T185)) → fc129_out_ga(T185, X419)
fc129_in_ga(T206, X511) → U73_ga(T206, X511, lessc71_in_g(T206))
U73_ga(T206, X511, lessc71_out_g(T206)) → U74_ga(T206, X511, addc102_in_ga(T206, T210))
U74_ga(T206, X511, addc102_out_ga(T206, T210)) → U75_ga(T206, X511, fc129_in_ga(T210, T213))
U75_ga(T206, X511, fc129_out_ga(T210, T213)) → U76_ga(T206, X511, fc129_in_ga(T213, X511))
U76_ga(T206, X511, fc129_out_ga(T213, X511)) → fc129_out_ga(T206, X511)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lessc10_in_g(x1)  =  lessc10_in_g(x1)
U42_g(x1, x2)  =  U42_g(x1, x2)
lessc16_in_g(x1)  =  lessc16_in_g(x1)
0  =  0
lessc16_out_g(x1)  =  lessc16_out_g(x1)
U41_g(x1, x2)  =  U41_g(x1, x2)
lessc10_out_g(x1)  =  lessc10_out_g(x1)
U43_g(x1, x2)  =  U43_g(x1, x2)
lessc59_in_g(x1)  =  lessc59_in_g(x1)
U49_g(x1, x2)  =  U49_g(x1, x2)
lessc59_out_g(x1)  =  lessc59_out_g(x1)
U50_g(x1, x2)  =  U50_g(x1, x2)
lessc71_in_g(x1)  =  lessc71_in_g(x1)
U69_g(x1, x2)  =  U69_g(x1, x2)
lessc77_in_g(x1)  =  lessc77_in_g(x1)
U51_g(x1, x2)  =  U51_g(x1, x2)
lessc83_in_g(x1)  =  lessc83_in_g(x1)
lessc83_out_g(x1)  =  lessc83_out_g(x1)
lessc77_out_g(x1)  =  lessc77_out_g(x1)
U52_g(x1, x2)  =  U52_g(x1, x2)
lessc71_out_g(x1)  =  lessc71_out_g(x1)
U70_g(x1, x2)  =  U70_g(x1, x2)
addc102_in_ga(x1, x2)  =  addc102_in_ga(x1)
U53_ga(x1, x2, x3)  =  U53_ga(x1, x3)
U54_ga(x1, x2, x3)  =  U54_ga(x1, x3)
addc110_in_ga(x1, x2)  =  addc110_in_ga(x1)
U55_ga(x1, x2, x3)  =  U55_ga(x1, x3)
U56_ga(x1, x2, x3)  =  U56_ga(x1, x3)
addc117_in_ga(x1, x2)  =  addc117_in_ga(x1)
addc117_out_ga(x1, x2)  =  addc117_out_ga(x1, x2)
U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)
addc110_out_ga(x1, x2)  =  addc110_out_ga(x1, x2)
addc102_out_ga(x1, x2)  =  addc102_out_ga(x1, x2)
fc129_in_ga(x1, x2)  =  fc129_in_ga(x1)
U71_ga(x1, x2, x3)  =  U71_ga(x1, x3)
U72_ga(x1, x2, x3)  =  U72_ga(x1, x3)
addc136_in_ag(x1, x2)  =  addc136_in_ag(x2)
U66_ag(x1, x2, x3)  =  U66_ag(x2, x3)
U67_ag(x1, x2, x3)  =  U67_ag(x2, x3)
addc143_in_ag(x1, x2)  =  addc143_in_ag(x2)
addc143_out_ag(x1, x2)  =  addc143_out_ag(x1, x2)
U68_ag(x1, x2, x3)  =  U68_ag(x2, x3)
addc136_out_ag(x1, x2)  =  addc136_out_ag(x1, x2)
fc129_out_ga(x1, x2)  =  fc129_out_ga(x1, x2)
U73_ga(x1, x2, x3)  =  U73_ga(x1, x3)
U74_ga(x1, x2, x3)  =  U74_ga(x1, x3)
U75_ga(x1, x2, x3)  =  U75_ga(x1, x3)
U76_ga(x1, x2, x3)  =  U76_ga(x1, x3)
F1_IN_GA(x1, x2)  =  F1_IN_GA(x1)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)

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