Left Termination of the query pattern average_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

average(0, 0, Z) :- ','(!, eq(Z, 0)).
average(0, s(0), Z) :- ','(!, eq(Z, 0)).
average(0, s(s(0)), Z) :- ','(!, eq(Z, s(0))).
average(X, Y, Z) :- ','(p(X, P), average(P, s(Y), Z)).
average(X, Y, Z) :- ','(p(Y, P1), ','(p(P1, P2), ','(p(P2, P3), ','(average(s(X), P3, U), p(Z, U))))).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

average(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: average(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: average(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T5, 0)) | average(0, 0, T3), SCOPE: 1, CLAUSE: 1 | average(0, 0, T3), SCOPE: 1, CLAUSE: 2 | average(0, 0, T3), SCOPE: 1, CLAUSE: 3 | average(0, 0, T3), SCOPE: 1, CLAUSE: 4 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: average(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | ?_1\n\nT1 is ground\nT2 is ground\nX2 is free; \naverage(T1, T2, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T5, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T5, 0), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(!_1, eq(T8, 0)) | average(0, s(0), T3), SCOPE: 1, CLAUSE: 2 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 4 | ?_1\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | ?_1\n\nT1 is ground\nT2 is ground\nX2 is free; \nX6 is free; \naverage(T1, T2, T3) !=? average(0, 0, X2)\naverage(T1, T2, T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T8, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: eq(T8, 0), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(!_1, eq(T11, s(0))) | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 4 | ?_1\n\nX2 is free\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nX2 is free; \nX6 is free; \nX10 is free; \naverage(T1, T2, T3) !=? average(0, 0, X2)\naverage(T1, T2, T3) !=? average(0, s(0), X6)\naverage(T1, T2, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: eq(T11, s(0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: eq(T11, s(0)), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX2 is free; \nX6 is free; \nX10 is free; \naverage(T1, T2, T3) !=? average(0, 0, X2)\naverage(T1, T2, T3) !=? average(0, s(0), X6)\naverage(T1, T2, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nX2 is free; \nX6 is free; \nX10 is free; \naverage(T1, T2, T3) !=? average(0, 0, X2)\naverage(T1, T2, T3) !=? average(0, s(0), X6)\naverage(T1, T2, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T13, X22), average(X22, s(T14), T16))\n\nT13 is ground\nT14 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX22 is free; \naverage(T13, T14, T3) !=? average(0, 0, X2)\naverage(T13, T14, T3) !=? average(0, s(0), X6)\naverage(T13, T14, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(p(T13, X22), average(X22, s(T14), T16)), SCOPE: 5, CLAUSE: 5 | ','(p(T13, X22), average(X22, s(T14), T16)), SCOPE: 5, CLAUSE: 6\n\nT13 is ground\nT14 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX22 is free; \naverage(T13, T14, T3) !=? average(0, 0, X2)\naverage(T13, T14, T3) !=? average(0, s(0), X6)\naverage(T13, T14, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: average(0, s(T14), T16) | ','(p(0, X22), average(X22, s(T14), T16)), SCOPE: 5, CLAUSE: 6\n\nT14 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX22 is free; \naverage(0, T14, T3) !=? average(0, 0, X2)\naverage(0, T14, T3) !=? average(0, s(0), X6)\naverage(0, T14, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T13, X22), average(X22, s(T14), T16)), SCOPE: 5, CLAUSE: 6\n\nT13 is ground\nT14 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX22 is free; \naverage(T13, T14, T3) !=? average(0, 0, X2)\naverage(T13, T14, T3) !=? average(0, s(0), X6)\naverage(T13, T14, T3) !=? average(0, s(s(0)), X10)\np(T13, X22) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: average(0, s(T14), T16)\n\nT14 is ground\nX2 is free; \nX6 is free; \nX10 is free; \naverage(0, T14, T3) !=? average(0, 0, X2)\naverage(0, T14, T3) !=? average(0, s(0), X6)\naverage(0, T14, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(0, X22), average(X22, s(T14), T16)), SCOPE: 5, CLAUSE: 6\n\nT14 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX22 is free; \naverage(0, T14, T3) !=? average(0, 0, X2)\naverage(0, T14, T3) !=? average(0, s(0), X6)\naverage(0, T14, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\nX2 is free\nX6 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: average(T17, s(T14), T16)\n\nT14 is ground\nT17 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\nX2 is free\nX6 is free; \nX10 is free; \nX22 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T19, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))))\n\nT18 is ground\nT19 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, T19, T3) !=? average(0, 0, X2)\naverage(T18, T19, T3) !=? average(0, s(0), X6)\naverage(T18, T19, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T19, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 5 | ','(p(T19, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nT19 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, T19, T3) !=? average(0, 0, X2)\naverage(T18, T19, T3) !=? average(0, s(0), X6)\naverage(T18, T19, T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))) | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(T19, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nT19 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, T19, T3) !=? average(0, 0, X2)\naverage(T18, T19, T3) !=? average(0, s(0), X6)\naverage(T18, T19, T3) !=? average(0, s(s(0)), X10)\np(T19, X32) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 5 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 6 | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))) | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 6 | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 8, CLAUSE: 5 | ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 8, CLAUSE: 6 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 6 | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(average(s(T18), 0, X35), p(T21, X35)) | ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 8, CLAUSE: 6 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 6 | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(average(s(T18), 0, X35), p(T21, X35))\n\nT18 is ground\nX2 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 8, CLAUSE: 6 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 6 | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: average(s(T18), 0, X35)\n\nT18 is ground\nX2 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(T21, T22)\n\nT22 is ground\nX2 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: p(T21, T22), SCOPE: 9, CLAUSE: 5 | p(T21, T22), SCOPE: 9, CLAUSE: 6\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true | p(T21, 0), SCOPE: 9, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: p(T21, T22), SCOPE: 9, CLAUSE: 6\n\nT22 is ground\np(T21, T22) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: p(T21, 0), SCOPE: 9, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 7, CLAUSE: 6 | ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(0, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(p(T25, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35))))\n\nT18 is ground\nT25 is ground\nX2 is free; \nX6 is free; \nX10 is free; \nX32 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(T25), T3) !=? average(0, s(0), X6)\naverage(T18, s(T25), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\nX2 is free\nX6 is free; \nX10 is free; \nX32 is free; \nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(T25, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 5 | ','(p(T25, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nT25 is ground\nX6 is free; \nX10 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(T25), T3) !=? average(0, s(0), X6)\naverage(T18, s(T25), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))) | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nX6 is free; \nX10 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(p(T25, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nT25 is ground\nX6 is free; \nX10 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(T25), T3) !=? average(0, s(0), X6)\naverage(T18, s(T25), T3) !=? average(0, s(s(0)), X10)\np(T25, X33) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 11, CLAUSE: 5 | ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 11, CLAUSE: 6 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nX6 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(average(s(T18), 0, X35), p(T21, X35)) | ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 11, CLAUSE: 6 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nX6 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(average(s(T18), 0, X35), p(T21, X35))\n\nT18 is ground\nX6 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 11, CLAUSE: 6 | ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nX6 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: average(s(T18), 0, X35)\n\nT18 is ground\nX6 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: p(T21, T26)\n\nT26 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(T21, T26), SCOPE: 12, CLAUSE: 5 | p(T21, T26), SCOPE: 12, CLAUSE: 6\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true | p(T21, 0), SCOPE: 12, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: p(T21, T26), SCOPE: 12, CLAUSE: 6\n\nT26 is ground\np(T21, T26) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: p(T21, 0), SCOPE: 12, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(0, X33), ','(p(X33, X34), ','(average(s(T18), X34, X35), p(T21, X35)))), SCOPE: 10, CLAUSE: 6\n\nT18 is ground\nX6 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(0), T3) !=? average(0, s(0), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(p(T29, X34), ','(average(s(T18), X34, X35), p(T21, X35)))\n\nT18 is ground\nT29 is ground\nX6 is free; \nX10 is free; \nX33 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(s(T29)), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\nX6 is free\nX10 is free; \nX33 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(p(T29, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 13, CLAUSE: 5 | ','(p(T29, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 13, CLAUSE: 6\n\nT18 is ground\nT29 is ground\nX10 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(s(T29)), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(average(s(T18), 0, X35), p(T21, X35)) | ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 13, CLAUSE: 6\n\nT18 is ground\nX10 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(s(0)), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(p(T29, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 13, CLAUSE: 6\n\nT18 is ground\nT29 is ground\nX10 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(s(T29)), T3) !=? average(0, s(s(0)), X10)\np(T29, X34) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(average(s(T18), 0, X35), p(T21, X35))\n\nT18 is ground\nX10 is free; \nX35 is free; \naverage(T18, s(s(0)), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(p(0, X34), ','(average(s(T18), X34, X35), p(T21, X35))), SCOPE: 13, CLAUSE: 6\n\nT18 is ground\nX10 is free; \nX34 is free; \nX35 is free; \naverage(T18, s(s(0)), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: average(s(T18), 0, X35)\n\nT18 is ground\nX10 is free; \nX35 is free; \naverage(T18, s(s(0)), T3) !=? average(0, s(s(0)), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: p(T21, T30)\n\nT30 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: p(T21, T30), SCOPE: 14, CLAUSE: 5 | p(T21, T30), SCOPE: 14, CLAUSE: 6\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: true | p(T21, 0), SCOPE: 14, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: p(T21, T30), SCOPE: 14, CLAUSE: 6\n\nT30 is ground\np(T21, T30) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: p(T21, 0), SCOPE: 14, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(average(s(T18), T33, X35), p(T21, X35))\n\nT18 is ground\nT33 is ground\nX10 is free; \nX34 is free; \nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\nX10 is free\nX34 is free; \nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: average(s(T18), T33, X35)\n\nT18 is ground\nT33 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: p(T21, T34)\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: p(T21, T34), SCOPE: 15, CLAUSE: 5 | p(T21, T34), SCOPE: 15, CLAUSE: 6\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: true | p(T21, 0), SCOPE: 15, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: p(T21, T34), SCOPE: 15, CLAUSE: 6\n\nT34 is ground\np(T21, T34) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: p(T21, 0), SCOPE: 15, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 117 [arrowhead = none ];
117 -> 2;

118 [label="EVAL with clause\naverage(0, 0, X2) :- ','(!, eq(X2, 0)).\nand substitution\nT1 -> 0\nT2 -> 0\nT3 -> T5\nX2 -> T5\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 118 [arrowhead = none ];
118 -> 3;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 119 [arrowhead = none ];
119 -> 4;

120 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 120 [arrowhead = none ];
120 -> 5;

121 [label="EVAL with clause\naverage(0, s(0), X6) :- ','(!, eq(X6, 0)).\nand substitution\nT1 -> 0\nT2 -> s(0)\nT3 -> T8\nX6 -> T8\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 121 [arrowhead = none ];
121 -> 10;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 122 [arrowhead = none ];
122 -> 11;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 123 [arrowhead = none ];
123 -> 6;

124 [label="EVAL with clause\neq(X4, X4).\nand substitution\nT5 -> 0\nX4 -> 0\nT6 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 124 [arrowhead = none ];
124 -> 7;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 125 [arrowhead = none ];
125 -> 8;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 126 [arrowhead = none ];
126 -> 9;

127 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 127 [arrowhead = none ];
127 -> 12;

128 [label="EVAL with clause\naverage(0, s(s(0)), X10) :- ','(!, eq(X10, s(0))).\nand substitution\nT1 -> 0\nT2 -> s(s(0))\nT3 -> T11\nX10 -> T11\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 128 [arrowhead = none ];
128 -> 17;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 129 [arrowhead = none ];
129 -> 18;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 130 [arrowhead = none ];
130 -> 13;

131 [label="EVAL with clause\neq(X8, X8).\nand substitution\nT8 -> 0\nX8 -> 0\nT9 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 131 [arrowhead = none ];
131 -> 14;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 132 [arrowhead = none ];
132 -> 15;

133 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 133 [arrowhead = none ];
133 -> 16;

134 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 134 [arrowhead = none ];
134 -> 19;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 135 [arrowhead = none ];
135 -> 24;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 136 [arrowhead = none ];
136 -> 25;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 137 [arrowhead = none ];
137 -> 20;

138 [label="EVAL with clause\neq(X12, X12).\nand substitution\nT11 -> s(0)\nX12 -> s(0)\nT12 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 138 [arrowhead = none ];
138 -> 21;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 139 [arrowhead = none ];
139 -> 22;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 140 [arrowhead = none ];
140 -> 23;

141 [label="EVAL with clause\naverage(X19, X20, X21) :- ','(p(X19, X22), average(X22, s(X20), X21)).\nand substitution\nT1 -> T13\nX19 -> T13\nT2 -> T14\nX20 -> T14\nT3 -> T16\nX21 -> T16\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 141 [arrowhead = none ];
141 -> 26;

142 [label="EVAL with clause\naverage(X29, X30, X31) :- ','(p(X30, X32), ','(p(X32, X33), ','(p(X33, X34), ','(average(s(X29), X34, X35), p(X31, X35))))).\nand substitution\nT1 -> T18\nX29 -> T18\nT2 -> T19\nX30 -> T19\nT3 -> T21\nX31 -> T21\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 142 [arrowhead = none ];
142 -> 35;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 143 [arrowhead = none ];
143 -> 27;

144 [label="EVAL with clause\np(0, 0).\nand substitution\nT13 -> 0\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 144 [arrowhead = none ];
144 -> 28;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 145 [arrowhead = none ];
145 -> 29;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 146 [arrowhead = none ];
146 -> 30;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 147 [arrowhead = none ];
147 -> 31;

148 [label="EVAL with clause\np(s(X25), X25).\nand substitution\nX25 -> T17\nT13 -> s(T17)\nX22 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 148 [arrowhead = none ];
148 -> 33;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 149 [arrowhead = none ];
149 -> 34;

150 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> s(T14)\nT3 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 150 [arrowhead = none , style=dashed];
150 -> 1[style=dashed];

151 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 151 [arrowhead = none ];
151 -> 32;

152 [label="INSTANCE with matching:\nT1 -> T17\nT2 -> s(T14)\nT3 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 152 [arrowhead = none , style=dashed];
152 -> 1[style=dashed];

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 153 [arrowhead = none ];
153 -> 36;

154 [label="EVAL with clause\np(0, 0).\nand substitution\nT19 -> 0\nX32 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 154 [arrowhead = none ];
154 -> 37;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 155 [arrowhead = none ];
155 -> 38;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 156 [arrowhead = none ];
156 -> 39;

157 [label="EVAL with clause\np(s(X44), X44).\nand substitution\nX44 -> T25\nT19 -> s(T25)\nX32 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 157 [arrowhead = none ];
157 -> 60;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 158 [arrowhead = none ];
158 -> 61;

159 [label="EVAL with clause\np(0, 0).\nand substitution\nX33 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 159 [arrowhead = none ];
159 -> 40;

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 160 [arrowhead = none ];
160 -> 41;

161 [label="EVAL with clause\np(0, 0).\nand substitution\nX34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 161 [arrowhead = none ];
161 -> 42;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 162 [arrowhead = none ];
162 -> 43;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 163 [arrowhead = none ];
163 -> 44;

164 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 164 [arrowhead = none ];
164 -> 45;

165 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX35 -> T22", fontsize=14, style = filled, fillcolor = violet];
43 -> 165 [arrowhead = none ];
165 -> 46;

166 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 166 [arrowhead = none ];
166 -> 57;

167 [label="INSTANCE with matching:\nT1 -> s(T18)\nT2 -> 0\nT3 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 167 [arrowhead = none , style=dashed];
167 -> 1[style=dashed];

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 168 [arrowhead = none ];
168 -> 47;

169 [label="EVAL with clause\np(0, 0).\nand substitution\nT21 -> 0\nT22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 169 [arrowhead = none ];
169 -> 48;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 170 [arrowhead = none ];
170 -> 49;

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

172 [label="EVAL with clause\np(s(X39), X39).\nand substitution\nX39 -> T24\nT21 -> s(T24)\nT22 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 172 [arrowhead = none ];
172 -> 54;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 173 [arrowhead = none ];
173 -> 55;

174 [label="EVAL with clause\np(s(X37), X37).\nand substitution\nX37 -> 0\nT21 -> s(0)\nT23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 174 [arrowhead = none ];
174 -> 51;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 175 [arrowhead = none ];
175 -> 52;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 176 [arrowhead = none ];
176 -> 53;

177 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 177 [arrowhead = none ];
177 -> 56;

178 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 178 [arrowhead = none ];
178 -> 58;

179 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
58 -> 179 [arrowhead = none ];
179 -> 59;

180 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 180 [arrowhead = none ];
180 -> 62;

181 [label="EVAL with clause\np(0, 0).\nand substitution\nT25 -> 0\nX33 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 181 [arrowhead = none ];
181 -> 63;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 182 [arrowhead = none ];
182 -> 64;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 183 [arrowhead = none ];
183 -> 65;

184 [label="EVAL with clause\np(s(X52), X52).\nand substitution\nX52 -> T29\nT25 -> s(T29)\nX33 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 184 [arrowhead = none ];
184 -> 83;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 185 [arrowhead = none ];
185 -> 84;

186 [label="EVAL with clause\np(0, 0).\nand substitution\nX34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 186 [arrowhead = none ];
186 -> 66;

187 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 187 [arrowhead = none ];
187 -> 67;

188 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 188 [arrowhead = none ];
188 -> 68;

189 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
67 -> 189 [arrowhead = none ];
189 -> 69;

190 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:\nX35 -> T26", fontsize=14, style = filled, fillcolor = violet];
67 -> 190 [arrowhead = none ];
190 -> 70;

191 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
68 -> 191 [arrowhead = none ];
191 -> 81;

192 [label="INSTANCE with matching:\nT1 -> s(T18)\nT2 -> 0\nT3 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 192 [arrowhead = none , style=dashed];
192 -> 1[style=dashed];

193 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 193 [arrowhead = none ];
193 -> 71;

194 [label="EVAL with clause\np(0, 0).\nand substitution\nT21 -> 0\nT26 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 194 [arrowhead = none ];
194 -> 72;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 195 [arrowhead = none ];
195 -> 73;

196 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 196 [arrowhead = none ];
196 -> 74;

197 [label="EVAL with clause\np(s(X48), X48).\nand substitution\nX48 -> T28\nT21 -> s(T28)\nT26 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 197 [arrowhead = none ];
197 -> 78;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 198 [arrowhead = none ];
198 -> 79;

199 [label="EVAL with clause\np(s(X46), X46).\nand substitution\nX46 -> 0\nT21 -> s(0)\nT27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 199 [arrowhead = none ];
199 -> 75;

200 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 200 [arrowhead = none ];
200 -> 76;

201 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 201 [arrowhead = none ];
201 -> 77;

202 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 202 [arrowhead = none ];
202 -> 80;

203 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
81 -> 203 [arrowhead = none ];
203 -> 82;

204 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 204 [arrowhead = none ];
204 -> 85;

205 [label="EVAL with clause\np(0, 0).\nand substitution\nT29 -> 0\nX34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 205 [arrowhead = none ];
205 -> 86;

206 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 206 [arrowhead = none ];
206 -> 87;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 207 [arrowhead = none ];
207 -> 88;

208 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 208 [arrowhead = none ];
208 -> 89;

209 [label="EVAL with clause\np(s(X59), X59).\nand substitution\nX59 -> T33\nT29 -> s(T33)\nX34 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 209 [arrowhead = none ];
209 -> 103;

210 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 210 [arrowhead = none ];
210 -> 104;

211 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
88 -> 211 [arrowhead = none ];
211 -> 90;

212 [label="SPLIT 2\nnew knowledge:\nT30 is ground\nreplacements:\nX35 -> T30", fontsize=14, style = filled, fillcolor = violet];
88 -> 212 [arrowhead = none ];
212 -> 91;

213 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
89 -> 213 [arrowhead = none ];
213 -> 102;

214 [label="INSTANCE with matching:\nT1 -> s(T18)\nT2 -> 0\nT3 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 214 [arrowhead = none , style=dashed];
214 -> 1[style=dashed];

215 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 215 [arrowhead = none ];
215 -> 92;

216 [label="EVAL with clause\np(0, 0).\nand substitution\nT21 -> 0\nT30 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 216 [arrowhead = none ];
216 -> 93;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 217 [arrowhead = none ];
217 -> 94;

218 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 218 [arrowhead = none ];
218 -> 95;

219 [label="EVAL with clause\np(s(X56), X56).\nand substitution\nX56 -> T32\nT21 -> s(T32)\nT30 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 219 [arrowhead = none ];
219 -> 99;

220 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 220 [arrowhead = none ];
220 -> 100;

221 [label="EVAL with clause\np(s(X54), X54).\nand substitution\nX54 -> 0\nT21 -> s(0)\nT31 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 221 [arrowhead = none ];
221 -> 96;

222 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 222 [arrowhead = none ];
222 -> 97;

223 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 223 [arrowhead = none ];
223 -> 98;

224 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 224 [arrowhead = none ];
224 -> 101;

225 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
103 -> 225 [arrowhead = none ];
225 -> 105;

226 [label="SPLIT 2\nnew knowledge:\nT34 is ground\nreplacements:\nX35 -> T34", fontsize=14, style = filled, fillcolor = violet];
103 -> 226 [arrowhead = none ];
226 -> 106;

227 [label="INSTANCE with matching:\nT1 -> s(T18)\nT2 -> T33\nT3 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 227 [arrowhead = none , style=dashed];
227 -> 1[style=dashed];

228 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
106 -> 228 [arrowhead = none ];
228 -> 107;

229 [label="EVAL with clause\np(0, 0).\nand substitution\nT21 -> 0\nT34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 229 [arrowhead = none ];
229 -> 108;

230 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 230 [arrowhead = none ];
230 -> 109;

231 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 231 [arrowhead = none ];
231 -> 110;

232 [label="EVAL with clause\np(s(X63), X63).\nand substitution\nX63 -> T36\nT21 -> s(T36)\nT34 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 232 [arrowhead = none ];
232 -> 114;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 233 [arrowhead = none ];
233 -> 115;

234 [label="EVAL with clause\np(s(X61), X61).\nand substitution\nX61 -> 0\nT21 -> s(0)\nT35 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 234 [arrowhead = none ];
234 -> 111;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 235 [arrowhead = none ];
235 -> 112;

236 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
111 -> 236 [arrowhead = none ];
236 -> 113;

237 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 237 [arrowhead = none ];
237 -> 116;

}

(2) Obligation:

Clauses:

average1(0, 0, 0).
average1(0, s(0), 0).
average1(0, s(s(0)), s(0)).
average1(0, T14, T16) :- average1(0, s(T14), T16).
average1(s(T17), T14, T16) :- average1(T17, s(T14), T16).
average1(T18, 0, T21) :- average1(s(T18), 0, X35).
average1(T18, 0, 0) :- average1(s(T18), 0, 0).
average1(T18, 0, s(0)) :- average1(s(T18), 0, 0).
average1(T18, 0, s(T24)) :- average1(s(T18), 0, T24).
average1(T18, s(0), T21) :- average1(s(T18), 0, X35).
average1(T18, s(0), 0) :- average1(s(T18), 0, 0).
average1(T18, s(0), s(0)) :- average1(s(T18), 0, 0).
average1(T18, s(0), s(T28)) :- average1(s(T18), 0, T28).
average1(T18, s(s(0)), T21) :- average1(s(T18), 0, X35).
average1(T18, s(s(0)), 0) :- average1(s(T18), 0, 0).
average1(T18, s(s(0)), s(0)) :- average1(s(T18), 0, 0).
average1(T18, s(s(0)), s(T32)) :- average1(s(T18), 0, T32).
average1(T18, s(s(s(T33))), T21) :- average1(s(T18), T33, X35).
average1(T18, s(s(s(T33))), 0) :- average1(s(T18), T33, 0).
average1(T18, s(s(s(T33))), s(0)) :- average1(s(T18), T33, 0).
average1(T18, s(s(s(T33))), s(T36)) :- average1(s(T18), T33, T36).

Queries:

average1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
average1_in: (b,b,f) (b,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3)  =  U3_gga(x1, x3)
U4_gga(x1, x2)  =  U4_gga(x1, x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3)  =  U1_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x1, x2, x3)
U5_gga(x1, x2)  =  U5_gga(x1, x2)
U4_ggg(x1, x2)  =  U4_ggg(x1, x2)
U5_ggg(x1, x2)  =  U5_ggg(x1, x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x1, x2, x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x1, x2, x3)
U6_gga(x1, x2, x3)  =  U6_gga(x1, x3)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x3)
U8_gga(x1, x2)  =  U8_gga(x1, x2)
U8_ggg(x1, x2)  =  U8_ggg(x1, x2)
U9_ggg(x1, x2)  =  U9_ggg(x1, x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U9_gga(x1, x2)  =  U9_gga(x1, x2)
U12_ggg(x1, x2)  =  U12_ggg(x1, x2)
U13_ggg(x1, x2)  =  U13_ggg(x1, x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x1, x2, x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x1, x2, x3, x4)
U10_gga(x1, x2, x3)  =  U10_gga(x1, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x3)
U12_gga(x1, x2)  =  U12_gga(x1, x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x1, x2, x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x1, x2, x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x1, x2, x3, x4)
U13_gga(x1, x2)  =  U13_gga(x1, x2)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x2, x3)
U17_gga(x1, x2, x3)  =  U17_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3)  =  U3_gga(x1, x3)
U4_gga(x1, x2)  =  U4_gga(x1, x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3)  =  U1_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x1, x2, x3)
U5_gga(x1, x2)  =  U5_gga(x1, x2)
U4_ggg(x1, x2)  =  U4_ggg(x1, x2)
U5_ggg(x1, x2)  =  U5_ggg(x1, x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x1, x2, x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x1, x2, x3)
U6_gga(x1, x2, x3)  =  U6_gga(x1, x3)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x3)
U8_gga(x1, x2)  =  U8_gga(x1, x2)
U8_ggg(x1, x2)  =  U8_ggg(x1, x2)
U9_ggg(x1, x2)  =  U9_ggg(x1, x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U9_gga(x1, x2)  =  U9_gga(x1, x2)
U12_ggg(x1, x2)  =  U12_ggg(x1, x2)
U13_ggg(x1, x2)  =  U13_ggg(x1, x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x1, x2, x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x1, x2, x3, x4)
U10_gga(x1, x2, x3)  =  U10_gga(x1, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x3)
U12_gga(x1, x2)  =  U12_gga(x1, x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x1, x2, x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x1, x2, x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x1, x2, x3, x4)
U13_gga(x1, x2)  =  U13_gga(x1, x2)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x2, x3)
U17_gga(x1, x2, x3)  =  U17_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x1, x2, x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

AVERAGE1_IN_GGA(0, T14, T16) → U1_GGA(T14, T16, average1_in_gga(0, s(T14), T16))
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(s(T17), T14, T16) → U2_GGA(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(T18, 0, T21) → U3_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → U4_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(0, T14, T16) → U1_GGG(T14, T16, average1_in_ggg(0, s(T14), T16))
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(s(T17), T14, T16) → U2_GGG(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(T18, 0, T21) → U3_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → U5_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → U4_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(0)) → U5_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → U6_GGG(T18, T24, average1_in_ggg(s(T18), 0, T24))
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), T21) → U7_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → U6_GGA(T18, T24, average1_in_gga(s(T18), 0, T24))
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGA(T18, s(0), T21) → U7_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), 0) → U8_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → U8_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → U9_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → U10_GGG(T18, T28, average1_in_ggg(s(T18), 0, T28))
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → U11_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → U9_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → U12_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → U13_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → U14_GGG(T18, T32, average1_in_ggg(s(T18), 0, T32))
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → U15_GGG(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → U10_GGA(T18, T28, average1_in_gga(s(T18), 0, T28))
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → U11_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → U12_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → U16_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → U17_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → U18_GGG(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → U13_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → U14_GGA(T18, T32, average1_in_gga(s(T18), 0, T32))
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → U15_GGA(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → U16_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → U17_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → U18_GGA(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)

The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3)  =  U3_gga(x1, x3)
U4_gga(x1, x2)  =  U4_gga(x1, x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3)  =  U1_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x1, x2, x3)
U5_gga(x1, x2)  =  U5_gga(x1, x2)
U4_ggg(x1, x2)  =  U4_ggg(x1, x2)
U5_ggg(x1, x2)  =  U5_ggg(x1, x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x1, x2, x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x1, x2, x3)
U6_gga(x1, x2, x3)  =  U6_gga(x1, x3)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x3)
U8_gga(x1, x2)  =  U8_gga(x1, x2)
U8_ggg(x1, x2)  =  U8_ggg(x1, x2)
U9_ggg(x1, x2)  =  U9_ggg(x1, x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U9_gga(x1, x2)  =  U9_gga(x1, x2)
U12_ggg(x1, x2)  =  U12_ggg(x1, x2)
U13_ggg(x1, x2)  =  U13_ggg(x1, x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x1, x2, x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x1, x2, x3, x4)
U10_gga(x1, x2, x3)  =  U10_gga(x1, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x3)
U12_gga(x1, x2)  =  U12_gga(x1, x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x1, x2, x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x1, x2, x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x1, x2, x3, x4)
U13_gga(x1, x2)  =  U13_gga(x1, x2)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x2, x3)
U17_gga(x1, x2, x3)  =  U17_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x1, x2, x4)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3)  =  U3_GGA(x1, x3)
U4_GGA(x1, x2)  =  U4_GGA(x1, x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3)  =  U1_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
U3_GGG(x1, x2, x3)  =  U3_GGG(x1, x2, x3)
U5_GGA(x1, x2)  =  U5_GGA(x1, x2)
U4_GGG(x1, x2)  =  U4_GGG(x1, x2)
U5_GGG(x1, x2)  =  U5_GGG(x1, x2)
U6_GGG(x1, x2, x3)  =  U6_GGG(x1, x2, x3)
U7_GGG(x1, x2, x3)  =  U7_GGG(x1, x2, x3)
U6_GGA(x1, x2, x3)  =  U6_GGA(x1, x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x1, x3)
U8_GGA(x1, x2)  =  U8_GGA(x1, x2)
U8_GGG(x1, x2)  =  U8_GGG(x1, x2)
U9_GGG(x1, x2)  =  U9_GGG(x1, x2)
U10_GGG(x1, x2, x3)  =  U10_GGG(x1, x2, x3)
U11_GGG(x1, x2, x3)  =  U11_GGG(x1, x2, x3)
U9_GGA(x1, x2)  =  U9_GGA(x1, x2)
U12_GGG(x1, x2)  =  U12_GGG(x1, x2)
U13_GGG(x1, x2)  =  U13_GGG(x1, x2)
U14_GGG(x1, x2, x3)  =  U14_GGG(x1, x2, x3)
U15_GGG(x1, x2, x3, x4)  =  U15_GGG(x1, x2, x3, x4)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x1, x3)
U12_GGA(x1, x2)  =  U12_GGA(x1, x2)
U16_GGG(x1, x2, x3)  =  U16_GGG(x1, x2, x3)
U17_GGG(x1, x2, x3)  =  U17_GGG(x1, x2, x3)
U18_GGG(x1, x2, x3, x4)  =  U18_GGG(x1, x2, x3, x4)
U13_GGA(x1, x2)  =  U13_GGA(x1, x2)
U14_GGA(x1, x2, x3)  =  U14_GGA(x1, x3)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3)  =  U16_GGA(x1, x2, x3)
U17_GGA(x1, x2, x3)  =  U17_GGA(x1, x2, x3)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x1, x2, x4)

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

(6) Obligation:

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

AVERAGE1_IN_GGA(0, T14, T16) → U1_GGA(T14, T16, average1_in_gga(0, s(T14), T16))
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(s(T17), T14, T16) → U2_GGA(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(T18, 0, T21) → U3_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → U4_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(0, T14, T16) → U1_GGG(T14, T16, average1_in_ggg(0, s(T14), T16))
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(s(T17), T14, T16) → U2_GGG(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(T18, 0, T21) → U3_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → U5_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → U4_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(0)) → U5_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → U6_GGG(T18, T24, average1_in_ggg(s(T18), 0, T24))
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), T21) → U7_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → U6_GGA(T18, T24, average1_in_gga(s(T18), 0, T24))
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGA(T18, s(0), T21) → U7_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), 0) → U8_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → U8_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → U9_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → U10_GGG(T18, T28, average1_in_ggg(s(T18), 0, T28))
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → U11_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → U9_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → U12_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → U13_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → U14_GGG(T18, T32, average1_in_ggg(s(T18), 0, T32))
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → U15_GGG(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → U10_GGA(T18, T28, average1_in_gga(s(T18), 0, T28))
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → U11_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → U12_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → U16_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → U17_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → U18_GGG(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → U13_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → U14_GGA(T18, T32, average1_in_gga(s(T18), 0, T32))
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → U15_GGA(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → U16_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → U17_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → U18_GGA(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)

The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3)  =  U3_gga(x1, x3)
U4_gga(x1, x2)  =  U4_gga(x1, x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3)  =  U1_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x1, x2, x3)
U5_gga(x1, x2)  =  U5_gga(x1, x2)
U4_ggg(x1, x2)  =  U4_ggg(x1, x2)
U5_ggg(x1, x2)  =  U5_ggg(x1, x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x1, x2, x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x1, x2, x3)
U6_gga(x1, x2, x3)  =  U6_gga(x1, x3)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x3)
U8_gga(x1, x2)  =  U8_gga(x1, x2)
U8_ggg(x1, x2)  =  U8_ggg(x1, x2)
U9_ggg(x1, x2)  =  U9_ggg(x1, x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U9_gga(x1, x2)  =  U9_gga(x1, x2)
U12_ggg(x1, x2)  =  U12_ggg(x1, x2)
U13_ggg(x1, x2)  =  U13_ggg(x1, x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x1, x2, x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x1, x2, x3, x4)
U10_gga(x1, x2, x3)  =  U10_gga(x1, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x3)
U12_gga(x1, x2)  =  U12_gga(x1, x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x1, x2, x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x1, x2, x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x1, x2, x3, x4)
U13_gga(x1, x2)  =  U13_gga(x1, x2)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x2, x3)
U17_gga(x1, x2, x3)  =  U17_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x1, x2, x4)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3)  =  U3_GGA(x1, x3)
U4_GGA(x1, x2)  =  U4_GGA(x1, x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3)  =  U1_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
U3_GGG(x1, x2, x3)  =  U3_GGG(x1, x2, x3)
U5_GGA(x1, x2)  =  U5_GGA(x1, x2)
U4_GGG(x1, x2)  =  U4_GGG(x1, x2)
U5_GGG(x1, x2)  =  U5_GGG(x1, x2)
U6_GGG(x1, x2, x3)  =  U6_GGG(x1, x2, x3)
U7_GGG(x1, x2, x3)  =  U7_GGG(x1, x2, x3)
U6_GGA(x1, x2, x3)  =  U6_GGA(x1, x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x1, x3)
U8_GGA(x1, x2)  =  U8_GGA(x1, x2)
U8_GGG(x1, x2)  =  U8_GGG(x1, x2)
U9_GGG(x1, x2)  =  U9_GGG(x1, x2)
U10_GGG(x1, x2, x3)  =  U10_GGG(x1, x2, x3)
U11_GGG(x1, x2, x3)  =  U11_GGG(x1, x2, x3)
U9_GGA(x1, x2)  =  U9_GGA(x1, x2)
U12_GGG(x1, x2)  =  U12_GGG(x1, x2)
U13_GGG(x1, x2)  =  U13_GGG(x1, x2)
U14_GGG(x1, x2, x3)  =  U14_GGG(x1, x2, x3)
U15_GGG(x1, x2, x3, x4)  =  U15_GGG(x1, x2, x3, x4)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x1, x3)
U12_GGA(x1, x2)  =  U12_GGA(x1, x2)
U16_GGG(x1, x2, x3)  =  U16_GGG(x1, x2, x3)
U17_GGG(x1, x2, x3)  =  U17_GGG(x1, x2, x3)
U18_GGG(x1, x2, x3, x4)  =  U18_GGG(x1, x2, x3, x4)
U13_GGA(x1, x2)  =  U13_GGA(x1, x2)
U14_GGA(x1, x2, x3)  =  U14_GGA(x1, x3)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3)  =  U16_GGA(x1, x2, x3)
U17_GGA(x1, x2, x3)  =  U17_GGA(x1, x2, x3)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x1, x2, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 36 less nodes.

(8) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3)  =  U3_gga(x1, x3)
U4_gga(x1, x2)  =  U4_gga(x1, x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3)  =  U1_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x1, x2, x3)
U5_gga(x1, x2)  =  U5_gga(x1, x2)
U4_ggg(x1, x2)  =  U4_ggg(x1, x2)
U5_ggg(x1, x2)  =  U5_ggg(x1, x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x1, x2, x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x1, x2, x3)
U6_gga(x1, x2, x3)  =  U6_gga(x1, x3)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x3)
U8_gga(x1, x2)  =  U8_gga(x1, x2)
U8_ggg(x1, x2)  =  U8_ggg(x1, x2)
U9_ggg(x1, x2)  =  U9_ggg(x1, x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U9_gga(x1, x2)  =  U9_gga(x1, x2)
U12_ggg(x1, x2)  =  U12_ggg(x1, x2)
U13_ggg(x1, x2)  =  U13_ggg(x1, x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x1, x2, x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x1, x2, x3, x4)
U10_gga(x1, x2, x3)  =  U10_gga(x1, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x3)
U12_gga(x1, x2)  =  U12_gga(x1, x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x1, x2, x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x1, x2, x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x1, x2, x3, x4)
U13_gga(x1, x2)  =  U13_gga(x1, x2)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x2, x3)
U17_gga(x1, x2, x3)  =  U17_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x1, x2, x4)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

R is empty.
The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

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

POL(0) = 0   
POL(AVERAGE1_IN_GGA(x₁, x₂)) = 0   
POL(AVERAGE1_IN_GGG(x₁, x₂, x₃)) = x₃   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)

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

(15) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)

(16) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)

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

(17) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14)) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))

(18) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))

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

(19) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)

(20) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)

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

(21) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)

(22) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)

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

(23) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)

(24) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)

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

(25) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)

(26) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)

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

(27) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)

(28) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)

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

(29) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)

(30) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)

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

(31) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)

(32) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)

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

(33) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)

(34) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)

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

(35) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(0, s(s(z0)), z1) → AVERAGE1_IN_GGG(0, s(s(s(z0))), z1)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)

(36) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)

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

(37) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(s(x0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(x0))), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)

(38) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)

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

(39) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)

(40) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)

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

(41) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14)) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))

(42) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))

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

(43) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)

(44) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)

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

(45) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)

(46) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)

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

(47) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(x0), 0, 0) → AVERAGE1_IN_GGG(x0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)

(48) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)

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

(49) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

(50) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

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

(51) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

(52) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

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

(53) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(x1), 0) → AVERAGE1_IN_GGG(0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(x1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)

(54) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)

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

(55) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(0), s(s(x1)), 0) → AVERAGE1_IN_GGG(0, s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(0)), x2) → AVERAGE1_IN_GGG(x0, s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)

(56) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(0)), x2) → AVERAGE1_IN_GGG(x0, s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)

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

(57) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(x0), s(s(0)), x2) → AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0)

(58) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)

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

(59) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), z2) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

(60) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

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

(61) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(s(x0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0)

(62) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

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

(63) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(s(s(x0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(x0)), s(s(s(0))), 0)

(64) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

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

(65) NonTerminationProof (EQUIVALENT transformation)

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

s = AVERAGE1_IN_GGA(0, s(z1)) evaluates to t =AVERAGE1_IN_GGA(0, s(s(z1)))

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

• Semiunifier: [ ]
• Matcher: [z1 / s(z1)]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from AVERAGE1_IN_GGA(0, s(z1)) to AVERAGE1_IN_GGA(0, s(s(z1))).

(66) FALSE

(67) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
average1_in: (b,b,f) (b,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3)  =  U3_gga(x3)
U4_gga(x1, x2)  =  U4_gga(x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg
U1_ggg(x1, x2, x3)  =  U1_ggg(x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x3)
U5_gga(x1, x2)  =  U5_gga(x2)
U4_ggg(x1, x2)  =  U4_ggg(x2)
U5_ggg(x1, x2)  =  U5_ggg(x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
U6_gga(x1, x2, x3)  =  U6_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
U8_gga(x1, x2)  =  U8_gga(x2)
U8_ggg(x1, x2)  =  U8_ggg(x2)
U9_ggg(x1, x2)  =  U9_ggg(x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x3)
U9_gga(x1, x2)  =  U9_gga(x2)
U12_ggg(x1, x2)  =  U12_ggg(x2)
U13_ggg(x1, x2)  =  U13_ggg(x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x4)
U10_gga(x1, x2, x3)  =  U10_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2)  =  U12_gga(x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x4)
U13_gga(x1, x2)  =  U13_gga(x2)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(68) Obligation:

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

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3)  =  U3_gga(x3)
U4_gga(x1, x2)  =  U4_gga(x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg
U1_ggg(x1, x2, x3)  =  U1_ggg(x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x3)
U5_gga(x1, x2)  =  U5_gga(x2)
U4_ggg(x1, x2)  =  U4_ggg(x2)
U5_ggg(x1, x2)  =  U5_ggg(x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
U6_gga(x1, x2, x3)  =  U6_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
U8_gga(x1, x2)  =  U8_gga(x2)
U8_ggg(x1, x2)  =  U8_ggg(x2)
U9_ggg(x1, x2)  =  U9_ggg(x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x3)
U9_gga(x1, x2)  =  U9_gga(x2)
U12_ggg(x1, x2)  =  U12_ggg(x2)
U13_ggg(x1, x2)  =  U13_ggg(x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x4)
U10_gga(x1, x2, x3)  =  U10_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2)  =  U12_gga(x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x4)
U13_gga(x1, x2)  =  U13_gga(x2)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x4)

(69) DependencyPairsProof (EQUIVALENT transformation)

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

AVERAGE1_IN_GGA(0, T14, T16) → U1_GGA(T14, T16, average1_in_gga(0, s(T14), T16))
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(s(T17), T14, T16) → U2_GGA(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(T18, 0, T21) → U3_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → U4_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(0, T14, T16) → U1_GGG(T14, T16, average1_in_ggg(0, s(T14), T16))
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(s(T17), T14, T16) → U2_GGG(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(T18, 0, T21) → U3_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → U5_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → U4_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(0)) → U5_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → U6_GGG(T18, T24, average1_in_ggg(s(T18), 0, T24))
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), T21) → U7_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → U6_GGA(T18, T24, average1_in_gga(s(T18), 0, T24))
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGA(T18, s(0), T21) → U7_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), 0) → U8_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → U8_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → U9_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → U10_GGG(T18, T28, average1_in_ggg(s(T18), 0, T28))
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → U11_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → U9_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → U12_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → U13_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → U14_GGG(T18, T32, average1_in_ggg(s(T18), 0, T32))
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → U15_GGG(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → U10_GGA(T18, T28, average1_in_gga(s(T18), 0, T28))
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → U11_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → U12_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → U16_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → U17_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → U18_GGG(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → U13_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → U14_GGA(T18, T32, average1_in_gga(s(T18), 0, T32))
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → U15_GGA(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → U16_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → U17_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → U18_GGA(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)

The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3)  =  U3_gga(x3)
U4_gga(x1, x2)  =  U4_gga(x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg
U1_ggg(x1, x2, x3)  =  U1_ggg(x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x3)
U5_gga(x1, x2)  =  U5_gga(x2)
U4_ggg(x1, x2)  =  U4_ggg(x2)
U5_ggg(x1, x2)  =  U5_ggg(x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
U6_gga(x1, x2, x3)  =  U6_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
U8_gga(x1, x2)  =  U8_gga(x2)
U8_ggg(x1, x2)  =  U8_ggg(x2)
U9_ggg(x1, x2)  =  U9_ggg(x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x3)
U9_gga(x1, x2)  =  U9_gga(x2)
U12_ggg(x1, x2)  =  U12_ggg(x2)
U13_ggg(x1, x2)  =  U13_ggg(x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x4)
U10_gga(x1, x2, x3)  =  U10_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2)  =  U12_gga(x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x4)
U13_gga(x1, x2)  =  U13_gga(x2)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x4)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GGA(x1, x2, x3)  =  U3_GGA(x3)
U4_GGA(x1, x2)  =  U4_GGA(x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3)  =  U1_GGG(x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x4)
U3_GGG(x1, x2, x3)  =  U3_GGG(x3)
U5_GGA(x1, x2)  =  U5_GGA(x2)
U4_GGG(x1, x2)  =  U4_GGG(x2)
U5_GGG(x1, x2)  =  U5_GGG(x2)
U6_GGG(x1, x2, x3)  =  U6_GGG(x3)
U7_GGG(x1, x2, x3)  =  U7_GGG(x3)
U6_GGA(x1, x2, x3)  =  U6_GGA(x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
U8_GGA(x1, x2)  =  U8_GGA(x2)
U8_GGG(x1, x2)  =  U8_GGG(x2)
U9_GGG(x1, x2)  =  U9_GGG(x2)
U10_GGG(x1, x2, x3)  =  U10_GGG(x3)
U11_GGG(x1, x2, x3)  =  U11_GGG(x3)
U9_GGA(x1, x2)  =  U9_GGA(x2)
U12_GGG(x1, x2)  =  U12_GGG(x2)
U13_GGG(x1, x2)  =  U13_GGG(x2)
U14_GGG(x1, x2, x3)  =  U14_GGG(x3)
U15_GGG(x1, x2, x3, x4)  =  U15_GGG(x4)
U10_GGA(x1, x2, x3)  =  U10_GGA(x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x3)
U12_GGA(x1, x2)  =  U12_GGA(x2)
U16_GGG(x1, x2, x3)  =  U16_GGG(x3)
U17_GGG(x1, x2, x3)  =  U17_GGG(x3)
U18_GGG(x1, x2, x3, x4)  =  U18_GGG(x4)
U13_GGA(x1, x2)  =  U13_GGA(x2)
U14_GGA(x1, x2, x3)  =  U14_GGA(x3)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x4)
U16_GGA(x1, x2, x3)  =  U16_GGA(x3)
U17_GGA(x1, x2, x3)  =  U17_GGA(x3)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x4)

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

(70) Obligation:

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

AVERAGE1_IN_GGA(0, T14, T16) → U1_GGA(T14, T16, average1_in_gga(0, s(T14), T16))
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(s(T17), T14, T16) → U2_GGA(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(T18, 0, T21) → U3_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → U4_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(0, T14, T16) → U1_GGG(T14, T16, average1_in_ggg(0, s(T14), T16))
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(s(T17), T14, T16) → U2_GGG(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(T18, 0, T21) → U3_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → U5_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → U4_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(0)) → U5_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → U6_GGG(T18, T24, average1_in_ggg(s(T18), 0, T24))
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), T21) → U7_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → U6_GGA(T18, T24, average1_in_gga(s(T18), 0, T24))
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGA(T18, s(0), T21) → U7_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), 0) → U8_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → U8_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → U9_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → U10_GGG(T18, T28, average1_in_ggg(s(T18), 0, T28))
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → U11_GGG(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → U9_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → U12_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → U13_GGG(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → U14_GGG(T18, T32, average1_in_ggg(s(T18), 0, T32))
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → U15_GGG(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → U10_GGA(T18, T28, average1_in_gga(s(T18), 0, T28))
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → U11_GGA(T18, T21, average1_in_gga(s(T18), 0, X35))
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → U12_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → U16_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → U17_GGG(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → U18_GGG(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → U13_GGA(T18, average1_in_ggg(s(T18), 0, 0))
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → U14_GGA(T18, T32, average1_in_gga(s(T18), 0, T32))
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → U15_GGA(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → U16_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → U17_GGA(T18, T33, average1_in_ggg(s(T18), T33, 0))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → U18_GGA(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)

The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3)  =  U3_gga(x3)
U4_gga(x1, x2)  =  U4_gga(x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg
U1_ggg(x1, x2, x3)  =  U1_ggg(x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x3)
U5_gga(x1, x2)  =  U5_gga(x2)
U4_ggg(x1, x2)  =  U4_ggg(x2)
U5_ggg(x1, x2)  =  U5_ggg(x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
U6_gga(x1, x2, x3)  =  U6_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
U8_gga(x1, x2)  =  U8_gga(x2)
U8_ggg(x1, x2)  =  U8_ggg(x2)
U9_ggg(x1, x2)  =  U9_ggg(x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x3)
U9_gga(x1, x2)  =  U9_gga(x2)
U12_ggg(x1, x2)  =  U12_ggg(x2)
U13_ggg(x1, x2)  =  U13_ggg(x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x4)
U10_gga(x1, x2, x3)  =  U10_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2)  =  U12_gga(x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x4)
U13_gga(x1, x2)  =  U13_gga(x2)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x4)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GGA(x1, x2, x3)  =  U3_GGA(x3)
U4_GGA(x1, x2)  =  U4_GGA(x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3)  =  U1_GGG(x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x4)
U3_GGG(x1, x2, x3)  =  U3_GGG(x3)
U5_GGA(x1, x2)  =  U5_GGA(x2)
U4_GGG(x1, x2)  =  U4_GGG(x2)
U5_GGG(x1, x2)  =  U5_GGG(x2)
U6_GGG(x1, x2, x3)  =  U6_GGG(x3)
U7_GGG(x1, x2, x3)  =  U7_GGG(x3)
U6_GGA(x1, x2, x3)  =  U6_GGA(x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
U8_GGA(x1, x2)  =  U8_GGA(x2)
U8_GGG(x1, x2)  =  U8_GGG(x2)
U9_GGG(x1, x2)  =  U9_GGG(x2)
U10_GGG(x1, x2, x3)  =  U10_GGG(x3)
U11_GGG(x1, x2, x3)  =  U11_GGG(x3)
U9_GGA(x1, x2)  =  U9_GGA(x2)
U12_GGG(x1, x2)  =  U12_GGG(x2)
U13_GGG(x1, x2)  =  U13_GGG(x2)
U14_GGG(x1, x2, x3)  =  U14_GGG(x3)
U15_GGG(x1, x2, x3, x4)  =  U15_GGG(x4)
U10_GGA(x1, x2, x3)  =  U10_GGA(x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x3)
U12_GGA(x1, x2)  =  U12_GGA(x2)
U16_GGG(x1, x2, x3)  =  U16_GGG(x3)
U17_GGG(x1, x2, x3)  =  U17_GGG(x3)
U18_GGG(x1, x2, x3, x4)  =  U18_GGG(x4)
U13_GGA(x1, x2)  =  U13_GGA(x2)
U14_GGA(x1, x2, x3)  =  U14_GGA(x3)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x4)
U16_GGA(x1, x2, x3)  =  U16_GGA(x3)
U17_GGA(x1, x2, x3)  =  U17_GGA(x3)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x4)

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

(71) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 36 less nodes.

(72) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

The TRS R consists of the following rules:

average1_in_gga(0, 0, 0) → average1_out_gga(0, 0, 0)
average1_in_gga(0, s(0), 0) → average1_out_gga(0, s(0), 0)
average1_in_gga(0, s(s(0)), s(0)) → average1_out_gga(0, s(s(0)), s(0))
average1_in_gga(0, T14, T16) → U1_gga(T14, T16, average1_in_gga(0, s(T14), T16))
average1_in_gga(s(T17), T14, T16) → U2_gga(T17, T14, T16, average1_in_gga(T17, s(T14), T16))
average1_in_gga(T18, 0, T21) → U3_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, 0) → U4_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(0, 0, 0) → average1_out_ggg(0, 0, 0)
average1_in_ggg(0, s(0), 0) → average1_out_ggg(0, s(0), 0)
average1_in_ggg(0, s(s(0)), s(0)) → average1_out_ggg(0, s(s(0)), s(0))
average1_in_ggg(0, T14, T16) → U1_ggg(T14, T16, average1_in_ggg(0, s(T14), T16))
average1_in_ggg(s(T17), T14, T16) → U2_ggg(T17, T14, T16, average1_in_ggg(T17, s(T14), T16))
average1_in_ggg(T18, 0, T21) → U3_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(0)) → U5_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, 0) → U4_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(0)) → U5_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, 0, s(T24)) → U6_ggg(T18, T24, average1_in_ggg(s(T18), 0, T24))
average1_in_ggg(T18, s(0), T21) → U7_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, 0, s(T24)) → U6_gga(T18, T24, average1_in_gga(s(T18), 0, T24))
average1_in_gga(T18, s(0), T21) → U7_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), 0) → U8_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), 0) → U8_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(0)) → U9_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(0), s(T28)) → U10_ggg(T18, T28, average1_in_ggg(s(T18), 0, T28))
average1_in_ggg(T18, s(s(0)), T21) → U11_ggg(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(0), s(0)) → U9_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), 0) → U12_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(0)) → U13_ggg(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(0)), s(T32)) → U14_ggg(T18, T32, average1_in_ggg(s(T18), 0, T32))
average1_in_ggg(T18, s(s(s(T33))), T21) → U15_ggg(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(0), s(T28)) → U10_gga(T18, T28, average1_in_gga(s(T18), 0, T28))
average1_in_gga(T18, s(s(0)), T21) → U11_gga(T18, T21, average1_in_gga(s(T18), 0, X35))
average1_in_gga(T18, s(s(0)), 0) → U12_gga(T18, average1_in_ggg(s(T18), 0, 0))
average1_in_ggg(T18, s(s(s(T33))), 0) → U16_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(0)) → U17_ggg(T18, T33, average1_in_ggg(s(T18), T33, 0))
average1_in_ggg(T18, s(s(s(T33))), s(T36)) → U18_ggg(T18, T33, T36, average1_in_ggg(s(T18), T33, T36))
U18_ggg(T18, T33, T36, average1_out_ggg(s(T18), T33, T36)) → average1_out_ggg(T18, s(s(s(T33))), s(T36))
U17_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), s(0))
U16_ggg(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_ggg(T18, s(s(s(T33))), 0)
U12_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), 0)
average1_in_gga(T18, s(s(0)), s(0)) → U13_gga(T18, average1_in_ggg(s(T18), 0, 0))
U13_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(s(0)), s(0))
average1_in_gga(T18, s(s(0)), s(T32)) → U14_gga(T18, T32, average1_in_gga(s(T18), 0, T32))
average1_in_gga(T18, s(s(s(T33))), T21) → U15_gga(T18, T33, T21, average1_in_gga(s(T18), T33, X35))
average1_in_gga(T18, s(s(s(T33))), 0) → U16_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U16_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), 0)
average1_in_gga(T18, s(s(s(T33))), s(0)) → U17_gga(T18, T33, average1_in_ggg(s(T18), T33, 0))
U17_gga(T18, T33, average1_out_ggg(s(T18), T33, 0)) → average1_out_gga(T18, s(s(s(T33))), s(0))
average1_in_gga(T18, s(s(s(T33))), s(T36)) → U18_gga(T18, T33, T36, average1_in_gga(s(T18), T33, T36))
U18_gga(T18, T33, T36, average1_out_gga(s(T18), T33, T36)) → average1_out_gga(T18, s(s(s(T33))), s(T36))
U15_gga(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_gga(T18, s(s(s(T33))), T21)
U14_gga(T18, T32, average1_out_gga(s(T18), 0, T32)) → average1_out_gga(T18, s(s(0)), s(T32))
U11_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(s(0)), T21)
U10_gga(T18, T28, average1_out_gga(s(T18), 0, T28)) → average1_out_gga(T18, s(0), s(T28))
U15_ggg(T18, T33, T21, average1_out_gga(s(T18), T33, X35)) → average1_out_ggg(T18, s(s(s(T33))), T21)
U14_ggg(T18, T32, average1_out_ggg(s(T18), 0, T32)) → average1_out_ggg(T18, s(s(0)), s(T32))
U13_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), s(0))
U12_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(s(0)), 0)
U9_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), s(0))
U11_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(s(0)), T21)
U10_ggg(T18, T28, average1_out_ggg(s(T18), 0, T28)) → average1_out_ggg(T18, s(0), s(T28))
U9_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), s(0))
U8_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, s(0), 0)
U8_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, s(0), 0)
U7_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, s(0), T21)
U6_gga(T18, T24, average1_out_gga(s(T18), 0, T24)) → average1_out_gga(T18, 0, s(T24))
U7_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, s(0), T21)
U6_ggg(T18, T24, average1_out_ggg(s(T18), 0, T24)) → average1_out_ggg(T18, 0, s(T24))
U5_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, s(0))
U4_ggg(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_ggg(T18, 0, 0)
U5_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, s(0))
U3_ggg(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_ggg(T18, 0, T21)
U2_ggg(T17, T14, T16, average1_out_ggg(T17, s(T14), T16)) → average1_out_ggg(s(T17), T14, T16)
U1_ggg(T14, T16, average1_out_ggg(0, s(T14), T16)) → average1_out_ggg(0, T14, T16)
U4_gga(T18, average1_out_ggg(s(T18), 0, 0)) → average1_out_gga(T18, 0, 0)
U3_gga(T18, T21, average1_out_gga(s(T18), 0, X35)) → average1_out_gga(T18, 0, T21)
U2_gga(T17, T14, T16, average1_out_gga(T17, s(T14), T16)) → average1_out_gga(s(T17), T14, T16)
U1_gga(T14, T16, average1_out_gga(0, s(T14), T16)) → average1_out_gga(0, T14, T16)

The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3)  =  average1_in_gga(x1, x2)
0  =  0
average1_out_gga(x1, x2, x3)  =  average1_out_gga
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3)  =  U3_gga(x3)
U4_gga(x1, x2)  =  U4_gga(x2)
average1_in_ggg(x1, x2, x3)  =  average1_in_ggg(x1, x2, x3)
average1_out_ggg(x1, x2, x3)  =  average1_out_ggg
U1_ggg(x1, x2, x3)  =  U1_ggg(x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x4)
U3_ggg(x1, x2, x3)  =  U3_ggg(x3)
U5_gga(x1, x2)  =  U5_gga(x2)
U4_ggg(x1, x2)  =  U4_ggg(x2)
U5_ggg(x1, x2)  =  U5_ggg(x2)
U6_ggg(x1, x2, x3)  =  U6_ggg(x3)
U7_ggg(x1, x2, x3)  =  U7_ggg(x3)
U6_gga(x1, x2, x3)  =  U6_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
U8_gga(x1, x2)  =  U8_gga(x2)
U8_ggg(x1, x2)  =  U8_ggg(x2)
U9_ggg(x1, x2)  =  U9_ggg(x2)
U10_ggg(x1, x2, x3)  =  U10_ggg(x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x3)
U9_gga(x1, x2)  =  U9_gga(x2)
U12_ggg(x1, x2)  =  U12_ggg(x2)
U13_ggg(x1, x2)  =  U13_ggg(x2)
U14_ggg(x1, x2, x3)  =  U14_ggg(x3)
U15_ggg(x1, x2, x3, x4)  =  U15_ggg(x4)
U10_gga(x1, x2, x3)  =  U10_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2)  =  U12_gga(x2)
U16_ggg(x1, x2, x3)  =  U16_ggg(x3)
U17_ggg(x1, x2, x3)  =  U17_ggg(x3)
U18_ggg(x1, x2, x3, x4)  =  U18_ggg(x4)
U13_gga(x1, x2)  =  U13_gga(x2)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3, x4)  =  U18_gga(x4)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)

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:

AVERAGE1_IN_GGA(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(s(T17), T14, T16) → AVERAGE1_IN_GGA(T17, s(T14), T16)
AVERAGE1_IN_GGA(0, T14, T16) → AVERAGE1_IN_GGA(0, s(T14), T16)
AVERAGE1_IN_GGA(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0, s(T24)) → AVERAGE1_IN_GGA(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(0), s(T28)) → AVERAGE1_IN_GGA(s(T18), 0, T28)
AVERAGE1_IN_GGA(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0, X35)
AVERAGE1_IN_GGA(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGA(s(T18), 0, T32)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33, X35)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGA(s(T18), T33, T36)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

R is empty.
The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
AVERAGE1_IN_GGA(x1, x2, x3)  =  AVERAGE1_IN_GGA(x1, x2)
AVERAGE1_IN_GGG(x1, x2, x3)  =  AVERAGE1_IN_GGG(x1, x2, x3)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AVERAGE1_IN_GGG(T18, s(0), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), s(T28)) → AVERAGE1_IN_GGG(s(T18), 0, T28)
AVERAGE1_IN_GGG(T18, 0, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, s(T24)) → AVERAGE1_IN_GGG(s(T18), 0, T24)
AVERAGE1_IN_GGG(T18, s(s(0)), s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), s(T32)) → AVERAGE1_IN_GGG(s(T18), 0, T32)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(0)) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), s(T36)) → AVERAGE1_IN_GGG(s(T18), T33, T36)

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

POL(0) = 0   
POL(AVERAGE1_IN_GGA(x₁, x₂)) = 0   
POL(AVERAGE1_IN_GGG(x₁, x₂, x₃)) = x₃   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)

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

(79) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, 0, T21) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)

(80) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)

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

(81) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(0, T14) → AVERAGE1_IN_GGA(0, s(T14)) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))

(82) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))

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

(83) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(T17), T14, T16) → AVERAGE1_IN_GGG(T17, s(T14), T16) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)

(84) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)

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

(85) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, T14, T16) → AVERAGE1_IN_GGG(0, s(T14), T16) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)

(86) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)

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

(87) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, s(0), T21) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)

(88) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)

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

(89) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)

(90) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)

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

(91) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(T18, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)

(92) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)

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

(93) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, 0, 0) → AVERAGE1_IN_GGG(s(T18), 0, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)

(94) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)

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

(95) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(x0), s(z1), z2) → AVERAGE1_IN_GGG(x0, s(s(z1)), z2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)

(96) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)

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

(97) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(T18, s(s(0)), T21) → AVERAGE1_IN_GGA(s(T18), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)

(98) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)

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

(99) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, s(z0), z1) → AVERAGE1_IN_GGG(0, s(s(z0)), z1) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(0, s(s(z0)), z1) → AVERAGE1_IN_GGG(0, s(s(s(z0))), z1)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)

(100) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)

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

(101) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, s(s(z1)), z2) → AVERAGE1_IN_GGG(0, s(s(s(z1))), z2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(s(x0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(x0))), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)

(102) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)

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

(103) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(0, s(s(0)), z1) → AVERAGE1_IN_GGA(s(0), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)

(104) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14))
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)

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

(105) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(s(T17), T14) → AVERAGE1_IN_GGA(T17, s(T14)) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))

(106) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))

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

(107) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(T18, s(s(s(T33))), T21) → AVERAGE1_IN_GGA(s(T18), T33) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)

(108) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)

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

(109) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGA(s(T18), T33) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)

(110) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)

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

(111) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(s(z0), z1, 0) → AVERAGE1_IN_GGG(z0, s(z1), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(x0), 0, 0) → AVERAGE1_IN_GGG(x0, s(0), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)

(112) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)

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

(113) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(T18, s(s(s(T33)))) → AVERAGE1_IN_GGG(s(T18), T33, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

(114) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

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

(115) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(T18, s(s(s(T33))), 0) → AVERAGE1_IN_GGG(s(T18), T33, 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

(116) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)

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

(117) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(s(z0), s(x1), 0) → AVERAGE1_IN_GGG(z0, s(s(x1)), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(x1), 0) → AVERAGE1_IN_GGG(0, s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(x1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)

(118) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)

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

(119) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGG(s(x0), s(s(z1)), z2) → AVERAGE1_IN_GGG(x0, s(s(s(z1))), z2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(0), s(s(x1)), 0) → AVERAGE1_IN_GGG(0, s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(0)), x2) → AVERAGE1_IN_GGG(x0, s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)

(120) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(0)), x2) → AVERAGE1_IN_GGG(x0, s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)

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

(121) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(x0), s(s(0)), x2) → AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0)

(122) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)

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

(123) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(x0, s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(x0), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), z2) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

(124) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

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

(125) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(s(x0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0)

(126) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

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

(127) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), x2) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGG(s(s(s(x0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(x0)), s(s(s(0))), 0)

(128) Obligation:

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

AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(0)) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(0), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(T18, s(s(0)), 0) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGA(s(T18), 0)
AVERAGE1_IN_GGA(T18, s(s(0))) → AVERAGE1_IN_GGG(s(T18), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(z0, s(0), 0)
AVERAGE1_IN_GGG(0, s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(z1), 0) → AVERAGE1_IN_GGG(0, s(s(z1)), 0)
AVERAGE1_IN_GGG(s(z0), s(0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(0), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGA(s(s(z0)), 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(z0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(z0), 0, 0) → AVERAGE1_IN_GGG(s(s(z0)), 0, 0)
AVERAGE1_IN_GGG(s(s(z0)), 0, 0) → AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(z0))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(s(z0)))), 0, 0) → AVERAGE1_IN_GGG(s(s(s(s(s(z0))))), 0, 0)
AVERAGE1_IN_GGG(s(x0), s(0), 0) → AVERAGE1_IN_GGG(x0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(z0), s(s(0)), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(z0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(0, s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(z0))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1)
AVERAGE1_IN_GGG(0, s(s(z0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(z0))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), z1) → AVERAGE1_IN_GGG(0, s(s(s(s(s(z0))))), z1)
AVERAGE1_IN_GGG(0, s(s(s(z0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), x2) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGG(0, s(s(s(x1))), x2) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(y_0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGA(s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)
AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), x1, 0) → AVERAGE1_IN_GGG(s(y_0), s(x1), 0)
AVERAGE1_IN_GGG(s(0), 0, 0) → AVERAGE1_IN_GGG(0, s(0), 0)
AVERAGE1_IN_GGG(s(0), x1, 0) → AVERAGE1_IN_GGG(0, s(x1), 0)
AVERAGE1_IN_GGG(s(s(y_0)), 0, 0) → AVERAGE1_IN_GGG(s(y_0), s(0), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(0), 0) → AVERAGE1_IN_GGG(0, s(s(0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(0), s(y_0), 0) → AVERAGE1_IN_GGG(0, s(s(y_0)), 0)
AVERAGE1_IN_GGG(s(0), s(s(0)), 0) → AVERAGE1_IN_GGG(0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(0)), 0) → AVERAGE1_IN_GGG(x0, s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGA(s(s(s(y_0))), s(s(s(0)))) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(0)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(0))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(s(y_1)))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(s(0))))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(s(y_1))))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGA(0, s(s(s(0)))) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(y_1))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(0))))) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(y_0)))))) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(y_0))))) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGA(0, s(s(s(s(s(s(y_0))))))) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGA(s(s(y_0)), s(s(s(s(s(0)))))) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(s(y_1)))))))) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGA(s(y_0), s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(x0), s(0), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(x0), 0, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), 0, 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), 0, 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(s(y_0)))), 0, 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(x0), s(y_1), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(y_1)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), x1, 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), 0, 0)
AVERAGE1_IN_GGG(0, s(s(s(x1))), 0) → AVERAGE1_IN_GGG(s(0), x1, 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(y_1), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(0), s(0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(y_0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(y_0), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(y_1)), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(x1), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(x1), 0) → AVERAGE1_IN_GGG(s(0), s(s(x1)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(0), 0) → AVERAGE1_IN_GGG(s(0), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(y_0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(y_1))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(0), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(0), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(0)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(0))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(0)))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(s(y_1)))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(s(y_1))))))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(0)))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0) → AVERAGE1_IN_GGG(0, s(s(s(s(s(s(y_0)))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(0))))), 0)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(x1)), x2) → AVERAGE1_IN_GGG(s(y_0), s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(x1)), x2) → AVERAGE1_IN_GGG(0, s(s(s(x1))), x2)
AVERAGE1_IN_GGG(s(0), s(s(s(y_0))), x2) → AVERAGE1_IN_GGG(0, s(s(s(s(y_0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(0))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(0)))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(0)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(0))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(s(y_1))))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(s(y_1)))))), x2)
AVERAGE1_IN_GGG(s(x0), s(s(s(s(y_1)))), x2) → AVERAGE1_IN_GGG(x0, s(s(s(s(s(y_1))))), x2)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(0))), s(s(x1)), 0) → AVERAGE1_IN_GGG(s(s(0)), s(s(s(x1))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(y_1))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(y_1)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(0)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(0))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(s(y_1))))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(s(y_1)))))))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(s(y_1)))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(s(y_1))))))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(0)), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(y_0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(y_0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(s(y_0)))), 0) → AVERAGE1_IN_GGG(s(0), s(s(s(s(s(y_0))))), 0)
AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(s(s(y_0)))), s(s(s(0))), 0) → AVERAGE1_IN_GGG(s(s(s(y_0))), s(s(s(s(0)))), 0)
AVERAGE1_IN_GGG(s(s(y_0)), s(s(s(s(s(y_1))))), 0) → AVERAGE1_IN_GGG(s(y_0), s(s(s(s(s(s(y_1)))))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(z0, s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(z0), 0)
AVERAGE1_IN_GGG(s(z0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGG(s(s(z0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(z0))), 0)
AVERAGE1_IN_GGG(s(0), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(0)), 0)
AVERAGE1_IN_GGG(s(s(s(z0))), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(s(z0)))), 0)
AVERAGE1_IN_GGG(0, s(s(s(0))), z1) → AVERAGE1_IN_GGA(s(0), 0)
AVERAGE1_IN_GGG(s(s(0)), s(s(s(0))), 0) → AVERAGE1_IN_GGA(s(s(s(0))), 0)

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

(129) NonTerminationProof (EQUIVALENT transformation)

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

s = AVERAGE1_IN_GGA(0, s(z1)) evaluates to t =AVERAGE1_IN_GGA(0, s(s(z1)))

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

• Semiunifier: [ ]
• Matcher: [z1 / s(z1)]

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Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from AVERAGE1_IN_GGA(0, s(z1)) to AVERAGE1_IN_GGA(0, s(s(z1))).

(130) FALSE