Left Termination of the query pattern max_valued_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

max_valued(.(Head, Tail), Max) :- max_valued(Tail, Head, Max).
max_valued([], Term, Term).
max_valued(.(Head, Tail), Term, Max) :- ','(higher_valued(Head, Term), ','(!, max_valued(Tail, Head, Max))).
max_valued(.(Head, Tail), Term, Max) :- ','(higher_valued(Term, Head), max_valued(Tail, Term, Max)).
higher_valued(X, Y) :- greater(s(X), Y).
greater(s(X1), 0).
greater(s(X), s(Y)) :- greater(X, Y).

Queries:

max_valued(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: max_valued(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: max_valued(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: max_valued(T4, T3, T6)\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\nX5 is free\nX6 is free; \nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 1 | max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 2 | max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 1\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 2 | max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(higher_valued(T8, T10), ','(!_2, max_valued(T9, T8, T12))) | max_valued(.(T8, T9), T10, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: max_valued(T4, T3, T6), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nT4 is ground\nX15 is free; \nX16 is free; \nX17 is free; \nX18 is free; \nmax_valued(T4, T3, T6) !=? max_valued(.(X15, X16), X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(higher_valued(T8, T10), ','(!_2, max_valued(T9, T8, T12))), SCOPE: 3, CLAUSE: 4 | max_valued(.(T8, T9), T10, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(greater(s(T13), T14), ','(!_2, max_valued(T9, T13, T12))) | max_valued(.(T13, T9), T14, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(greater(s(T13), T14), ','(!_2, max_valued(T9, T13, T12))), SCOPE: 4, CLAUSE: 5 | ','(greater(s(T13), T14), ','(!_2, max_valued(T9, T13, T12))), SCOPE: 4, CLAUSE: 6 | max_valued(.(T13, T9), T14, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(!_2, max_valued(T9, T15, T12)) | ','(greater(s(T15), 0), ','(!_2, max_valued(T9, T15, T12))), SCOPE: 4, CLAUSE: 6 | max_valued(.(T15, T9), 0, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(greater(s(T13), T14), ','(!_2, max_valued(T9, T13, T12))), SCOPE: 4, CLAUSE: 6 | max_valued(.(T13, T9), T14, T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT13 is ground\nT14 is ground\nX24 is free; \ngreater(s(T13), T14) !=? greater(s(X24), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: max_valued(T9, T15, T12)\n\nT9 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(greater(T16, T17), ','(!_2, max_valued(T9, T16, T12))) | max_valued(.(T16, T9), s(T17), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT16 is ground\nT17 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: max_valued(.(T13, T9), T14, T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT13 is ground\nT14 is ground\nX24 is free; \nX27 is free; \nX28 is free; \ngreater(s(T13), T14) !=? greater(s(X24), 0)\ngreater(s(T13), T14) !=? greater(s(X27), s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(greater(T16, T17), ','(!_2, max_valued(T9, T16, T12))), SCOPE: 5, CLAUSE: 5 | ','(greater(T16, T17), ','(!_2, max_valued(T9, T16, T12))), SCOPE: 5, CLAUSE: 6 | max_valued(.(T16, T9), s(T17), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT16 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(!_2, max_valued(T9, s(T18), T12)) | ','(greater(s(T18), 0), ','(!_2, max_valued(T9, s(T18), T12))), SCOPE: 5, CLAUSE: 6 | max_valued(.(s(T18), T9), s(0), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(greater(T16, T17), ','(!_2, max_valued(T9, T16, T12))), SCOPE: 5, CLAUSE: 6 | max_valued(.(T16, T9), s(T17), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT16 is ground\nT17 is ground\nX30 is free; \ngreater(T16, T17) !=? greater(s(X30), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: max_valued(T9, s(T18), T12)\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(greater(T19, T20), ','(!_2, max_valued(T9, s(T19), T12))) | max_valued(.(s(T19), T9), s(s(T20)), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT19 is ground\nT20 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: max_valued(.(T16, T9), s(T17), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT16 is ground\nT17 is ground\nX30 is free; \nX33 is free; \nX34 is free; \ngreater(T16, T17) !=? greater(s(X30), 0)\ngreater(T16, T17) !=? greater(s(X33), s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(greater(T19, T20), ','(!_2, max_valued(T9, s(T19), T12))), SCOPE: 6, CLAUSE: 5 | ','(greater(T19, T20), ','(!_2, max_valued(T9, s(T19), T12))), SCOPE: 6, CLAUSE: 6 | max_valued(.(s(T19), T9), s(s(T20)), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(!_2, max_valued(T9, s(s(T21)), T12)) | ','(greater(s(T21), 0), ','(!_2, max_valued(T9, s(s(T21)), T12))), SCOPE: 6, CLAUSE: 6 | max_valued(.(s(s(T21)), T9), s(s(0)), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(greater(T19, T20), ','(!_2, max_valued(T9, s(T19), T12))), SCOPE: 6, CLAUSE: 6 | max_valued(.(s(T19), T9), s(s(T20)), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT19 is ground\nT20 is ground\nX36 is free; \ngreater(T19, T20) !=? greater(s(X36), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: max_valued(T9, s(s(T21)), T12)\n\nT9 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(greater(T22, T23), ','(!_2, max_valued(T9, s(s(T22)), T12))) | max_valued(.(s(s(T22)), T9), s(s(s(T23))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT22 is ground\nT23 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: max_valued(.(s(T19), T9), s(s(T20)), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT19 is ground\nT20 is ground\nX36 is free; \nX39 is free; \nX40 is free; \ngreater(T19, T20) !=? greater(s(X36), 0)\ngreater(T19, T20) !=? greater(s(X39), s(X40))", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(greater(T22, T23), ','(!_2, max_valued(T9, s(s(T22)), T12))), SCOPE: 7, CLAUSE: 5 | ','(greater(T22, T23), ','(!_2, max_valued(T9, s(s(T22)), T12))), SCOPE: 7, CLAUSE: 6 | max_valued(.(s(s(T22)), T9), s(s(s(T23))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(!_2, max_valued(T9, s(s(s(T24))), T12)) | ','(greater(s(T24), 0), ','(!_2, max_valued(T9, s(s(s(T24))), T12))), SCOPE: 7, CLAUSE: 6 | max_valued(.(s(s(s(T24))), T9), s(s(s(0))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(greater(T22, T23), ','(!_2, max_valued(T9, s(s(T22)), T12))), SCOPE: 7, CLAUSE: 6 | max_valued(.(s(s(T22)), T9), s(s(s(T23))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT22 is ground\nT23 is ground\nX42 is free; \ngreater(T22, T23) !=? greater(s(X42), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: max_valued(T9, s(s(s(T24))), T12)\n\nT9 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(greater(T25, T26), ','(!_2, max_valued(T9, s(s(s(T25))), T12))) | max_valued(.(s(s(s(T25))), T9), s(s(s(s(T26)))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT25 is ground\nT26 is ground\nX42 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: max_valued(.(s(s(T22)), T9), s(s(s(T23))), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT22 is ground\nT23 is ground\nX42 is free; \nX45 is free; \nX46 is free; \ngreater(T22, T23) !=? greater(s(X42), 0)\ngreater(T22, T23) !=? greater(s(X45), s(X46))", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(greater(T25, T26), ','(!_2, max_valued(T9, s(s(s(T25))), T12))), SCOPE: 8, CLAUSE: 5 | ','(greater(T25, T26), ','(!_2, max_valued(T9, s(s(s(T25))), T12))), SCOPE: 8, CLAUSE: 6 | max_valued(.(s(s(s(T25))), T9), s(s(s(s(T26)))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(!_2, max_valued(T9, s(s(s(s(T27)))), T12)) | ','(greater(s(T27), 0), ','(!_2, max_valued(T9, s(s(s(s(T27)))), T12))), SCOPE: 8, CLAUSE: 6 | max_valued(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(greater(T25, T26), ','(!_2, max_valued(T9, s(s(s(T25))), T12))), SCOPE: 8, CLAUSE: 6 | max_valued(.(s(s(s(T25))), T9), s(s(s(s(T26)))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT25 is ground\nT26 is ground\nX48 is free; \ngreater(T25, T26) !=? greater(s(X48), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: max_valued(T9, s(s(s(s(T27)))), T12)\n\nT9 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(greater(T28, T29), ','(!_2, max_valued(T9, s(s(s(s(T28)))), T12))) | max_valued(.(s(s(s(s(T28)))), T9), s(s(s(s(s(T29))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT28 is ground\nT29 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: max_valued(.(s(s(s(T25))), T9), s(s(s(s(T26)))), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT25 is ground\nT26 is ground\nX48 is free; \nX51 is free; \nX52 is free; \ngreater(T25, T26) !=? greater(s(X48), 0)\ngreater(T25, T26) !=? greater(s(X51), s(X52))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(greater(T28, T29), ','(!_2, max_valued(T9, s(s(s(s(T28)))), T12))), SCOPE: 9, CLAUSE: 5 | ','(greater(T28, T29), ','(!_2, max_valued(T9, s(s(s(s(T28)))), T12))), SCOPE: 9, CLAUSE: 6 | max_valued(.(s(s(s(s(T28)))), T9), s(s(s(s(s(T29))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT28 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_2, max_valued(T9, s(s(s(s(s(T30))))), T12)) | ','(greater(s(T30), 0), ','(!_2, max_valued(T9, s(s(s(s(s(T30))))), T12))), SCOPE: 9, CLAUSE: 6 | max_valued(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(greater(T28, T29), ','(!_2, max_valued(T9, s(s(s(s(T28)))), T12))), SCOPE: 9, CLAUSE: 6 | max_valued(.(s(s(s(s(T28)))), T9), s(s(s(s(s(T29))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT28 is ground\nT29 is ground\nX54 is free; \ngreater(T28, T29) !=? greater(s(X54), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: max_valued(T9, s(s(s(s(s(T30))))), T12)\n\nT9 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(greater(T31, T32), ','(!_2, max_valued(T9, s(s(s(s(s(T31))))), T12))) | max_valued(.(s(s(s(s(s(T31))))), T9), s(s(s(s(s(s(T32)))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT31 is ground\nT32 is ground\nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: max_valued(.(s(s(s(s(T28)))), T9), s(s(s(s(s(T29))))), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT28 is ground\nT29 is ground\nX54 is free; \nX57 is free; \nX58 is free; \ngreater(T28, T29) !=? greater(s(X54), 0)\ngreater(T28, T29) !=? greater(s(X57), s(X58))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(greater(T31, T32), ','(!_2, max_valued(T9, s(s(s(s(s(T31))))), T12))), SCOPE: 10, CLAUSE: 5 | ','(greater(T31, T32), ','(!_2, max_valued(T9, s(s(s(s(s(T31))))), T12))), SCOPE: 10, CLAUSE: 6 | max_valued(.(s(s(s(s(s(T31))))), T9), s(s(s(s(s(s(T32)))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(!_2, max_valued(T9, s(s(s(s(s(s(T33)))))), T12)) | ','(greater(s(T33), 0), ','(!_2, max_valued(T9, s(s(s(s(s(s(T33)))))), T12))), SCOPE: 10, CLAUSE: 6 | max_valued(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(greater(T31, T32), ','(!_2, max_valued(T9, s(s(s(s(s(T31))))), T12))), SCOPE: 10, CLAUSE: 6 | max_valued(.(s(s(s(s(s(T31))))), T9), s(s(s(s(s(s(T32)))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT31 is ground\nT32 is ground\nX60 is free; \ngreater(T31, T32) !=? greater(s(X60), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: max_valued(T9, s(s(s(s(s(s(T33)))))), T12)\n\nT9 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(greater(T34, T35), ','(!_2, max_valued(T9, s(s(s(s(s(s(T34)))))), T12))) | max_valued(.(s(s(s(s(s(s(T34)))))), T9), s(s(s(s(s(s(s(T35))))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT34 is ground\nT35 is ground\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: max_valued(.(s(s(s(s(s(T31))))), T9), s(s(s(s(s(s(T32)))))), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT31 is ground\nT32 is ground\nX60 is free; \nX63 is free; \nX64 is free; \ngreater(T31, T32) !=? greater(s(X60), 0)\ngreater(T31, T32) !=? greater(s(X63), s(X64))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(greater(T34, T35), ','(!_2, max_valued(T9, s(s(s(s(s(s(T34)))))), T12))), SCOPE: 11, CLAUSE: 5 | ','(greater(T34, T35), ','(!_2, max_valued(T9, s(s(s(s(s(s(T34)))))), T12))), SCOPE: 11, CLAUSE: 6 | max_valued(.(s(s(s(s(s(s(T34)))))), T9), s(s(s(s(s(s(s(T35))))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(!_2, max_valued(T9, s(s(s(s(s(s(s(T36))))))), T12)) | ','(greater(s(T36), 0), ','(!_2, max_valued(T9, s(s(s(s(s(s(s(T36))))))), T12))), SCOPE: 11, CLAUSE: 6 | max_valued(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(greater(T34, T35), ','(!_2, max_valued(T9, s(s(s(s(s(s(T34)))))), T12))), SCOPE: 11, CLAUSE: 6 | max_valued(.(s(s(s(s(s(s(T34)))))), T9), s(s(s(s(s(s(s(T35))))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT34 is ground\nT35 is ground\nX66 is free; \ngreater(T34, T35) !=? greater(s(X66), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: max_valued(T9, s(s(s(s(s(s(s(T36))))))), T12)\n\nT9 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(greater(T37, T38), ','(!_2, max_valued(T9, s(s(s(s(s(s(s(T37))))))), T12))) | max_valued(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT9 is ground\nT37 is ground\nT38 is ground\nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: max_valued(.(s(s(s(s(s(s(T34)))))), T9), s(s(s(s(s(s(s(T35))))))), T6), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT34 is ground\nT35 is ground\nX66 is free; \nX69 is free; \nX70 is free; \ngreater(T34, T35) !=? greater(s(X66), 0)\ngreater(T34, T35) !=? greater(s(X69), s(X70))", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(greater(T37, T38), ','(!_2, max_valued(T9, s(T39), T12)))\n\nT9 is ground\nT37 is ground\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: greater(T37, T38)\n\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(!_2, max_valued(T9, s(T39), T12))\n\nT9 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: greater(T37, T38), SCOPE: 12, CLAUSE: 5 | greater(T37, T38), SCOPE: 12, CLAUSE: 6\n\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: greater(T37, T38), SCOPE: 12, CLAUSE: 5\n\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: greater(T37, T38), SCOPE: 12, CLAUSE: 6\n\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: greater(T41, T42)\n\nT41 is ground\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\nX76 is free\nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: max_valued(T9, s(T39), T12)\n\nT9 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(higher_valued(s(s(s(s(s(s(s(T45))))))), s(s(s(s(s(s(T43))))))), max_valued(T44, s(s(s(s(s(s(s(T45))))))), T47))\n\nT43 is ground\nT44 is ground\nT45 is ground\nX66 is free; \nX69 is free; \nX70 is free; \ngreater(T43, T45) !=? greater(s(X66), 0)\ngreater(T43, T45) !=? greater(s(X69), s(X70))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(higher_valued(s(T48), s(s(s(s(s(s(T43))))))), max_valued(T44, s(T49), T47))\n\nT43 is ground\nT44 is ground\nT48 is ground\nT49 is ground\nX66 is free; \nX69 is free; \nX70 is free; \ngreater(T43, T45) !=? greater(s(X66), 0)\ngreater(T43, T45) !=? greater(s(X69), s(X70))", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: higher_valued(s(T48), s(s(s(s(s(s(T43)))))))\n\nT43 is ground\nT48 is ground\nX66 is free; \nX69 is free; \nX70 is free; \ngreater(T43, T45) !=? greater(s(X66), 0)\ngreater(T43, T45) !=? greater(s(X69), s(X70))", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: max_valued(T44, s(T49), T47)\n\nT44 is ground\nT49 is ground\nX66 is free; \nX69 is free; \nX70 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: higher_valued(s(T48), s(s(s(s(s(s(T43))))))), SCOPE: 13, CLAUSE: 4\n\nT43 is ground\nT48 is ground\nX66 is free; \nX69 is free; \nX70 is free; \ngreater(T43, T45) !=? greater(s(X66), 0)\ngreater(T43, T45) !=? greater(s(X69), s(X70))", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: greater(s(s(T50)), s(s(s(s(s(s(T51)))))))\n\nT50 is ground\nT51 is ground\nX66 is free; \nX69 is free; \nX70 is free; \ngreater(T51, T45) !=? greater(s(X66), 0)\ngreater(T51, T45) !=? greater(s(X69), s(X70))", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(higher_valued(s(s(s(s(s(s(T54)))))), s(s(s(s(s(T52)))))), max_valued(T53, s(s(s(s(s(s(T54)))))), T56))\n\nT52 is ground\nT53 is ground\nT54 is ground\nX60 is free; \nX63 is free; \nX64 is free; \ngreater(T52, T54) !=? greater(s(X60), 0)\ngreater(T52, T54) !=? greater(s(X63), s(X64))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: higher_valued(s(s(s(s(s(s(T54)))))), s(s(s(s(s(T52))))))\n\nT52 is ground\nT54 is ground\nX60 is free; \nX63 is free; \nX64 is free; \ngreater(T52, T54) !=? greater(s(X60), 0)\ngreater(T52, T54) !=? greater(s(X63), s(X64))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: max_valued(T53, s(s(s(s(s(s(T54)))))), T56)\n\nT53 is ground\nT54 is ground\nX60 is free; \nX63 is free; \nX64 is free; \ngreater(T52, T54) !=? greater(s(X60), 0)\ngreater(T52, T54) !=? greater(s(X63), s(X64))", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: higher_valued(s(s(s(s(s(s(T54)))))), s(s(s(s(s(T52)))))), SCOPE: 14, CLAUSE: 4\n\nT52 is ground\nT54 is ground\nX60 is free; \nX63 is free; \nX64 is free; \ngreater(T52, T54) !=? greater(s(X60), 0)\ngreater(T52, T54) !=? greater(s(X63), s(X64))", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: greater(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))\n\nT57 is ground\nT58 is ground\nX60 is free; \nX63 is free; \nX64 is free; \ngreater(T58, T57) !=? greater(s(X60), 0)\ngreater(T58, T57) !=? greater(s(X63), s(X64))", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(higher_valued(s(s(s(s(s(T61))))), s(s(s(s(T59))))), max_valued(T60, s(s(s(s(s(T61))))), T63))\n\nT59 is ground\nT60 is ground\nT61 is ground\nX54 is free; \nX57 is free; \nX58 is free; \ngreater(T59, T61) !=? greater(s(X54), 0)\ngreater(T59, T61) !=? greater(s(X57), s(X58))", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: higher_valued(s(s(s(s(s(T61))))), s(s(s(s(T59)))))\n\nT59 is ground\nT61 is ground\nX54 is free; \nX57 is free; \nX58 is free; \ngreater(T59, T61) !=? greater(s(X54), 0)\ngreater(T59, T61) !=? greater(s(X57), s(X58))", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: max_valued(T60, s(s(s(s(s(T61))))), T63)\n\nT60 is ground\nT61 is ground\nX54 is free; \nX57 is free; \nX58 is free; \ngreater(T59, T61) !=? greater(s(X54), 0)\ngreater(T59, T61) !=? greater(s(X57), s(X58))", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: higher_valued(s(s(s(s(s(T61))))), s(s(s(s(T59))))), SCOPE: 15, CLAUSE: 4\n\nT59 is ground\nT61 is ground\nX54 is free; \nX57 is free; \nX58 is free; \ngreater(T59, T61) !=? greater(s(X54), 0)\ngreater(T59, T61) !=? greater(s(X57), s(X58))", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: greater(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))\n\nT64 is ground\nT65 is ground\nX54 is free; \nX57 is free; \nX58 is free; \ngreater(T65, T64) !=? greater(s(X54), 0)\ngreater(T65, T64) !=? greater(s(X57), s(X58))", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(higher_valued(s(s(s(s(T68)))), s(s(s(T66)))), max_valued(T67, s(s(s(s(T68)))), T70))\n\nT66 is ground\nT67 is ground\nT68 is ground\nX48 is free; \nX51 is free; \nX52 is free; \ngreater(T66, T68) !=? greater(s(X48), 0)\ngreater(T66, T68) !=? greater(s(X51), s(X52))", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: higher_valued(s(s(s(s(T68)))), s(s(s(T66))))\n\nT66 is ground\nT68 is ground\nX48 is free; \nX51 is free; \nX52 is free; \ngreater(T66, T68) !=? greater(s(X48), 0)\ngreater(T66, T68) !=? greater(s(X51), s(X52))", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: max_valued(T67, s(s(s(s(T68)))), T70)\n\nT67 is ground\nT68 is ground\nX48 is free; \nX51 is free; \nX52 is free; \ngreater(T66, T68) !=? greater(s(X48), 0)\ngreater(T66, T68) !=? greater(s(X51), s(X52))", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: higher_valued(s(s(s(s(T68)))), s(s(s(T66)))), SCOPE: 16, CLAUSE: 4\n\nT66 is ground\nT68 is ground\nX48 is free; \nX51 is free; \nX52 is free; \ngreater(T66, T68) !=? greater(s(X48), 0)\ngreater(T66, T68) !=? greater(s(X51), s(X52))", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: greater(s(s(s(s(s(T71))))), s(s(s(T72))))\n\nT71 is ground\nT72 is ground\nX48 is free; \nX51 is free; \nX52 is free; \ngreater(T72, T71) !=? greater(s(X48), 0)\ngreater(T72, T71) !=? greater(s(X51), s(X52))", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(higher_valued(s(s(s(T75))), s(s(T73))), max_valued(T74, s(s(s(T75))), T77))\n\nT73 is ground\nT74 is ground\nT75 is ground\nX42 is free; \nX45 is free; \nX46 is free; \ngreater(T73, T75) !=? greater(s(X42), 0)\ngreater(T73, T75) !=? greater(s(X45), s(X46))", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: higher_valued(s(s(s(T75))), s(s(T73)))\n\nT73 is ground\nT75 is ground\nX42 is free; \nX45 is free; \nX46 is free; \ngreater(T73, T75) !=? greater(s(X42), 0)\ngreater(T73, T75) !=? greater(s(X45), s(X46))", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: max_valued(T74, s(s(s(T75))), T77)\n\nT74 is ground\nT75 is ground\nX42 is free; \nX45 is free; \nX46 is free; \ngreater(T73, T75) !=? greater(s(X42), 0)\ngreater(T73, T75) !=? greater(s(X45), s(X46))", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: higher_valued(s(s(s(T75))), s(s(T73))), SCOPE: 17, CLAUSE: 4\n\nT73 is ground\nT75 is ground\nX42 is free; \nX45 is free; \nX46 is free; \ngreater(T73, T75) !=? greater(s(X42), 0)\ngreater(T73, T75) !=? greater(s(X45), s(X46))", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: greater(s(s(s(s(T78)))), s(s(T79)))\n\nT78 is ground\nT79 is ground\nX42 is free; \nX45 is free; \nX46 is free; \ngreater(T79, T78) !=? greater(s(X42), 0)\ngreater(T79, T78) !=? greater(s(X45), s(X46))", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(higher_valued(s(s(T82)), s(T80)), max_valued(T81, s(s(T82)), T84))\n\nT80 is ground\nT81 is ground\nT82 is ground\nX36 is free; \nX39 is free; \nX40 is free; \ngreater(T80, T82) !=? greater(s(X36), 0)\ngreater(T80, T82) !=? greater(s(X39), s(X40))", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: higher_valued(s(s(T82)), s(T80))\n\nT80 is ground\nT82 is ground\nX36 is free; \nX39 is free; \nX40 is free; \ngreater(T80, T82) !=? greater(s(X36), 0)\ngreater(T80, T82) !=? greater(s(X39), s(X40))", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: max_valued(T81, s(s(T82)), T84)\n\nT81 is ground\nT82 is ground\nX36 is free; \nX39 is free; \nX40 is free; \ngreater(T80, T82) !=? greater(s(X36), 0)\ngreater(T80, T82) !=? greater(s(X39), s(X40))", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: higher_valued(s(s(T82)), s(T80)), SCOPE: 18, CLAUSE: 4\n\nT80 is ground\nT82 is ground\nX36 is free; \nX39 is free; \nX40 is free; \ngreater(T80, T82) !=? greater(s(X36), 0)\ngreater(T80, T82) !=? greater(s(X39), s(X40))", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: greater(s(s(s(T85))), s(T86))\n\nT85 is ground\nT86 is ground\nX36 is free; \nX39 is free; \nX40 is free; \ngreater(T86, T85) !=? greater(s(X36), 0)\ngreater(T86, T85) !=? greater(s(X39), s(X40))", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(higher_valued(s(T89), T87), max_valued(T88, s(T89), T91))\n\nT87 is ground\nT88 is ground\nT89 is ground\nX30 is free; \nX33 is free; \nX34 is free; \ngreater(T87, T89) !=? greater(s(X30), 0)\ngreater(T87, T89) !=? greater(s(X33), s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: higher_valued(s(T89), T87)\n\nT87 is ground\nT89 is ground\nX30 is free; \nX33 is free; \nX34 is free; \ngreater(T87, T89) !=? greater(s(X30), 0)\ngreater(T87, T89) !=? greater(s(X33), s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: max_valued(T88, s(T89), T91)\n\nT88 is ground\nT89 is ground\nX30 is free; \nX33 is free; \nX34 is free; \ngreater(T87, T89) !=? greater(s(X30), 0)\ngreater(T87, T89) !=? greater(s(X33), s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: higher_valued(s(T89), T87), SCOPE: 19, CLAUSE: 4\n\nT87 is ground\nT89 is ground\nX30 is free; \nX33 is free; \nX34 is free; \ngreater(T87, T89) !=? greater(s(X30), 0)\ngreater(T87, T89) !=? greater(s(X33), s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: greater(s(s(T92)), T93)\n\nT92 is ground\nT93 is ground\nX30 is free; \nX33 is free; \nX34 is free; \ngreater(T93, T92) !=? greater(s(X30), 0)\ngreater(T93, T92) !=? greater(s(X33), s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(higher_valued(T96, T94), max_valued(T95, T96, T98))\n\nT94 is ground\nT95 is ground\nT96 is ground\nX24 is free; \nX27 is free; \nX28 is free; \ngreater(s(T94), T96) !=? greater(s(X24), 0)\ngreater(s(T94), T96) !=? greater(s(X27), s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: higher_valued(T96, T94)\n\nT94 is ground\nT96 is ground\nX24 is free; \nX27 is free; \nX28 is free; \ngreater(s(T94), T96) !=? greater(s(X24), 0)\ngreater(s(T94), T96) !=? greater(s(X27), s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: max_valued(T95, T96, T98)\n\nT95 is ground\nT96 is ground\nX24 is free; \nX27 is free; \nX28 is free; \ngreater(s(T94), T96) !=? greater(s(X24), 0)\ngreater(s(T94), T96) !=? greater(s(X27), s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: higher_valued(T96, T94), SCOPE: 20, CLAUSE: 4\n\nT94 is ground\nT96 is ground\nX24 is free; \nX27 is free; \nX28 is free; \ngreater(s(T94), T96) !=? greater(s(X24), 0)\ngreater(s(T94), T96) !=? greater(s(X27), s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: greater(s(T99), T100)\n\nT99 is ground\nT100 is ground\nX24 is free; \nX27 is free; \nX28 is free; \ngreater(s(T100), T99) !=? greater(s(X24), 0)\ngreater(s(T100), T99) !=? greater(s(X27), s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(higher_valued(T103, T101), max_valued(T102, T103, T105))\n\nT101 is ground\nT102 is ground\nT103 is ground\nX15 is free; \nX16 is free; \nX17 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\nX15 is free\nX16 is free; \nX17 is free; \nX18 is free; \nX178 is free; \nX179 is free; \nX180 is free; \nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: higher_valued(T103, T101)\n\nT101 is ground\nT103 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: max_valued(T102, T103, T105)\n\nT102 is ground\nT103 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: higher_valued(T103, T101), SCOPE: 21, CLAUSE: 4\n\nT101 is ground\nT103 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: greater(s(T106), T107)\n\nT106 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 122 [arrowhead = none ];
122 -> 2;

123 [label="EVAL with clause\nmax_valued(.(X5, X6), X7) :- max_valued(X6, X5, X7).\nand substitution\nX5 -> T3\nX6 -> T4\nT1 -> .(T3, T4)\nT2 -> T6\nX7 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 123 [arrowhead = none ];
123 -> 3;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 124 [arrowhead = none ];
124 -> 4;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 125 [arrowhead = none ];
125 -> 5;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 126 [arrowhead = none ];
126 -> 6;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 127 [arrowhead = none ];
127 -> 7;

128 [label="EVAL with clause\nmax_valued([], X10, X10).\nand substitution\nT4 -> []\nT3 -> T7\nX10 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 128 [arrowhead = none ];
128 -> 8;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 129 [arrowhead = none ];
129 -> 9;

130 [label="EVAL with clause\nmax_valued(.(X15, X16), X17, X18) :- ','(higher_valued(X15, X17), ','(!, max_valued(X16, X15, X18))).\nand substitution\nX15 -> T8\nX16 -> T9\nT4 -> .(T8, T9)\nT3 -> T10\nX17 -> T10\nT6 -> T12\nX18 -> T12\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 130 [arrowhead = none ];
130 -> 11;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 131 [arrowhead = none ];
131 -> 12;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 132 [arrowhead = none ];
132 -> 10;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 133 [arrowhead = none ];
133 -> 13;

134 [label="EVAL with clause\nmax_valued(.(X178, X179), X180, X181) :- ','(higher_valued(X180, X178), max_valued(X179, X180, X181)).\nand substitution\nX178 -> T101\nX179 -> T102\nT4 -> .(T101, T102)\nT3 -> T103\nX180 -> T103\nT6 -> T105\nX181 -> T105\nT104 -> T105", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 134 [arrowhead = none ];
134 -> 116;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 135 [arrowhead = none ];
135 -> 117;

136 [label="EVAL with clause\nhigher_valued(X21, X22) :- greater(s(X21), X22).\nand substitution\nT8 -> T13\nX21 -> T13\nT10 -> T14\nX22 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 136 [arrowhead = none ];
136 -> 14;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 137 [arrowhead = none ];
137 -> 15;

138 [label="EVAL with clause\ngreater(s(X24), 0).\nand substitution\nT13 -> T15\nX24 -> T15\nT14 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 138 [arrowhead = none ];
138 -> 16;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 139 [arrowhead = none ];
139 -> 17;

140 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 140 [arrowhead = none ];
140 -> 18;

141 [label="EVAL with clause\ngreater(s(X27), s(X28)) :- greater(X27, X28).\nand substitution\nT13 -> T16\nX27 -> T16\nX28 -> T17\nT14 -> s(T17)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 141 [arrowhead = none ];
141 -> 19;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 142 [arrowhead = none ];
142 -> 20;

143 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> T15\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 143 [arrowhead = none , style=dashed];
143 -> 3[style=dashed];

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 144 [arrowhead = none ];
144 -> 21;

145 [label="EVAL with clause\nmax_valued(.(X166, X167), X168, X169) :- ','(higher_valued(X168, X166), max_valued(X167, X168, X169)).\nand substitution\nT13 -> T94\nX166 -> T94\nT9 -> T95\nX167 -> T95\nT14 -> T96\nX168 -> T96\nT6 -> T98\nX169 -> T98\nT97 -> T98", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 145 [arrowhead = none ];
145 -> 111;

146 [label="EVAL with clause\ngreater(s(X30), 0).\nand substitution\nX30 -> T18\nT16 -> s(T18)\nT17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 146 [arrowhead = none ];
146 -> 22;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 147 [arrowhead = none ];
147 -> 23;

148 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 148 [arrowhead = none ];
148 -> 24;

149 [label="EVAL with clause\ngreater(s(X33), s(X34)) :- greater(X33, X34).\nand substitution\nX33 -> T19\nT16 -> s(T19)\nX34 -> T20\nT17 -> s(T20)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 149 [arrowhead = none ];
149 -> 25;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 150 [arrowhead = none ];
150 -> 26;

151 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(T18)\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 151 [arrowhead = none , style=dashed];
151 -> 3[style=dashed];

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 152 [arrowhead = none ];
152 -> 27;

153 [label="EVAL with clause\nmax_valued(.(X154, X155), X156, X157) :- ','(higher_valued(X156, X154), max_valued(X155, X156, X157)).\nand substitution\nT16 -> T87\nX154 -> T87\nT9 -> T88\nX155 -> T88\nT17 -> T89\nX156 -> s(T89)\nT6 -> T91\nX157 -> T91\nT90 -> T91", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 153 [arrowhead = none ];
153 -> 106;

154 [label="EVAL with clause\ngreater(s(X36), 0).\nand substitution\nX36 -> T21\nT19 -> s(T21)\nT20 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 154 [arrowhead = none ];
154 -> 28;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 155 [arrowhead = none ];
155 -> 29;

156 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 156 [arrowhead = none ];
156 -> 30;

157 [label="EVAL with clause\ngreater(s(X39), s(X40)) :- greater(X39, X40).\nand substitution\nX39 -> T22\nT19 -> s(T22)\nX40 -> T23\nT20 -> s(T23)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 157 [arrowhead = none ];
157 -> 31;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 158 [arrowhead = none ];
158 -> 32;

159 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(s(T21))\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 159 [arrowhead = none , style=dashed];
159 -> 3[style=dashed];

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 160 [arrowhead = none ];
160 -> 33;

161 [label="EVAL with clause\nmax_valued(.(X142, X143), X144, X145) :- ','(higher_valued(X144, X142), max_valued(X143, X144, X145)).\nand substitution\nT19 -> T80\nX142 -> s(T80)\nT9 -> T81\nX143 -> T81\nT20 -> T82\nX144 -> s(s(T82))\nT6 -> T84\nX145 -> T84\nT83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 161 [arrowhead = none ];
161 -> 101;

162 [label="EVAL with clause\ngreater(s(X42), 0).\nand substitution\nX42 -> T24\nT22 -> s(T24)\nT23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 162 [arrowhead = none ];
162 -> 34;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 163 [arrowhead = none ];
163 -> 35;

164 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 164 [arrowhead = none ];
164 -> 36;

165 [label="EVAL with clause\ngreater(s(X45), s(X46)) :- greater(X45, X46).\nand substitution\nX45 -> T25\nT22 -> s(T25)\nX46 -> T26\nT23 -> s(T26)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 165 [arrowhead = none ];
165 -> 37;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 166 [arrowhead = none ];
166 -> 38;

167 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(s(s(T24)))\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 167 [arrowhead = none , style=dashed];
167 -> 3[style=dashed];

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 168 [arrowhead = none ];
168 -> 39;

169 [label="EVAL with clause\nmax_valued(.(X130, X131), X132, X133) :- ','(higher_valued(X132, X130), max_valued(X131, X132, X133)).\nand substitution\nT22 -> T73\nX130 -> s(s(T73))\nT9 -> T74\nX131 -> T74\nT23 -> T75\nX132 -> s(s(s(T75)))\nT6 -> T77\nX133 -> T77\nT76 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 169 [arrowhead = none ];
169 -> 96;

170 [label="EVAL with clause\ngreater(s(X48), 0).\nand substitution\nX48 -> T27\nT25 -> s(T27)\nT26 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 170 [arrowhead = none ];
170 -> 40;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 171 [arrowhead = none ];
171 -> 41;

172 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 172 [arrowhead = none ];
172 -> 42;

173 [label="EVAL with clause\ngreater(s(X51), s(X52)) :- greater(X51, X52).\nand substitution\nX51 -> T28\nT25 -> s(T28)\nX52 -> T29\nT26 -> s(T29)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 173 [arrowhead = none ];
173 -> 43;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 174 [arrowhead = none ];
174 -> 44;

175 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(s(s(s(T27))))\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 175 [arrowhead = none , style=dashed];
175 -> 3[style=dashed];

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 176 [arrowhead = none ];
176 -> 45;

177 [label="EVAL with clause\nmax_valued(.(X118, X119), X120, X121) :- ','(higher_valued(X120, X118), max_valued(X119, X120, X121)).\nand substitution\nT25 -> T66\nX118 -> s(s(s(T66)))\nT9 -> T67\nX119 -> T67\nT26 -> T68\nX120 -> s(s(s(s(T68))))\nT6 -> T70\nX121 -> T70\nT69 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 177 [arrowhead = none ];
177 -> 91;

178 [label="EVAL with clause\ngreater(s(X54), 0).\nand substitution\nX54 -> T30\nT28 -> s(T30)\nT29 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 178 [arrowhead = none ];
178 -> 46;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 179 [arrowhead = none ];
179 -> 47;

180 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 180 [arrowhead = none ];
180 -> 48;

181 [label="EVAL with clause\ngreater(s(X57), s(X58)) :- greater(X57, X58).\nand substitution\nX57 -> T31\nT28 -> s(T31)\nX58 -> T32\nT29 -> s(T32)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 181 [arrowhead = none ];
181 -> 49;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 182 [arrowhead = none ];
182 -> 50;

183 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(s(s(s(s(T30)))))\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 183 [arrowhead = none , style=dashed];
183 -> 3[style=dashed];

184 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 184 [arrowhead = none ];
184 -> 51;

185 [label="EVAL with clause\nmax_valued(.(X106, X107), X108, X109) :- ','(higher_valued(X108, X106), max_valued(X107, X108, X109)).\nand substitution\nT28 -> T59\nX106 -> s(s(s(s(T59))))\nT9 -> T60\nX107 -> T60\nT29 -> T61\nX108 -> s(s(s(s(s(T61)))))\nT6 -> T63\nX109 -> T63\nT62 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 185 [arrowhead = none ];
185 -> 86;

186 [label="EVAL with clause\ngreater(s(X60), 0).\nand substitution\nX60 -> T33\nT31 -> s(T33)\nT32 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 186 [arrowhead = none ];
186 -> 52;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 187 [arrowhead = none ];
187 -> 53;

188 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 188 [arrowhead = none ];
188 -> 54;

189 [label="EVAL with clause\ngreater(s(X63), s(X64)) :- greater(X63, X64).\nand substitution\nX63 -> T34\nT31 -> s(T34)\nX64 -> T35\nT32 -> s(T35)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 189 [arrowhead = none ];
189 -> 55;

190 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 190 [arrowhead = none ];
190 -> 56;

191 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(s(s(s(s(s(T33))))))\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 191 [arrowhead = none , style=dashed];
191 -> 3[style=dashed];

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 192 [arrowhead = none ];
192 -> 57;

193 [label="EVAL with clause\nmax_valued(.(X94, X95), X96, X97) :- ','(higher_valued(X96, X94), max_valued(X95, X96, X97)).\nand substitution\nT31 -> T52\nX94 -> s(s(s(s(s(T52)))))\nT9 -> T53\nX95 -> T53\nT32 -> T54\nX96 -> s(s(s(s(s(s(T54))))))\nT6 -> T56\nX97 -> T56\nT55 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 193 [arrowhead = none ];
193 -> 81;

194 [label="EVAL with clause\ngreater(s(X66), 0).\nand substitution\nX66 -> T36\nT34 -> s(T36)\nT35 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 194 [arrowhead = none ];
194 -> 58;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 195 [arrowhead = none ];
195 -> 59;

196 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 196 [arrowhead = none ];
196 -> 60;

197 [label="EVAL with clause\ngreater(s(X69), s(X70)) :- greater(X69, X70).\nand substitution\nX69 -> T37\nT34 -> s(T37)\nX70 -> T38\nT35 -> s(T38)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 197 [arrowhead = none ];
197 -> 61;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 198 [arrowhead = none ];
198 -> 62;

199 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(s(s(s(s(s(s(T36)))))))\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 199 [arrowhead = none , style=dashed];
199 -> 3[style=dashed];

200 [label="GENERALIZATION\nT39 <-- s(s(s(s(s(s(T37))))))\n\nNew Knowledge:\nT39 is ground", fontsize=14, style = filled, fillcolor = violet];
61 -> 200 [arrowhead = none ];
200 -> 63;

201 [label="EVAL with clause\nmax_valued(.(X82, X83), X84, X85) :- ','(higher_valued(X84, X82), max_valued(X83, X84, X85)).\nand substitution\nT34 -> T43\nX82 -> s(s(s(s(s(s(T43))))))\nT9 -> T44\nX83 -> T44\nT35 -> T45\nX84 -> s(s(s(s(s(s(s(T45)))))))\nT6 -> T47\nX85 -> T47\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 201 [arrowhead = none ];
201 -> 75;

202 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
63 -> 202 [arrowhead = none ];
202 -> 64;

203 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
63 -> 203 [arrowhead = none ];
203 -> 65;

204 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 204 [arrowhead = none ];
204 -> 66;

205 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 205 [arrowhead = none ];
205 -> 74;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 206 [arrowhead = none ];
206 -> 67;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 207 [arrowhead = none ];
207 -> 68;

208 [label="EVAL with clause\ngreater(s(X73), 0).\nand substitution\nX73 -> T40\nT37 -> s(T40)\nT38 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 208 [arrowhead = none ];
208 -> 69;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 209 [arrowhead = none ];
209 -> 70;

210 [label="EVAL with clause\ngreater(s(X76), s(X77)) :- greater(X76, X77).\nand substitution\nX76 -> T41\nT37 -> s(T41)\nX77 -> T42\nT38 -> s(T42)", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 210 [arrowhead = none ];
210 -> 72;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 211 [arrowhead = none ];
211 -> 73;

212 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 212 [arrowhead = none ];
212 -> 71;

213 [label="INSTANCE with matching:\nT37 -> T41\nT38 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 213 [arrowhead = none , style=dashed];
213 -> 64[style=dashed];

214 [label="INSTANCE with matching:\nT4 -> T9\nT3 -> s(T39)\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 214 [arrowhead = none , style=dashed];
214 -> 3[style=dashed];

215 [label="GENERALIZATION\nT48 <-- s(s(s(s(s(s(T45))))))\nT49 <-- s(s(s(s(s(s(T45))))))\n\nNew Knowledge:\nT48 is ground\nT49 is ground", fontsize=14, style = filled, fillcolor = violet];
75 -> 215 [arrowhead = none ];
215 -> 76;

216 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
76 -> 216 [arrowhead = none ];
216 -> 77;

217 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
76 -> 217 [arrowhead = none ];
217 -> 78;

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 218 [arrowhead = none ];
218 -> 79;

219 [label="INSTANCE with matching:\nT4 -> T44\nT3 -> s(T49)\nT6 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 219 [arrowhead = none , style=dashed];
219 -> 3[style=dashed];

220 [label="EVAL with clause\nhigher_valued(X88, X89) :- greater(s(X88), X89).\nand substitution\nT48 -> T50\nX88 -> s(T50)\nT43 -> T51\nX89 -> s(s(s(s(s(s(T51))))))", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 220 [arrowhead = none ];
220 -> 80;

221 [label="INSTANCE with matching:\nT37 -> s(s(T50))\nT38 -> s(s(s(s(s(s(T51))))))", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 221 [arrowhead = none , style=dashed];
221 -> 64[style=dashed];

222 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
81 -> 222 [arrowhead = none ];
222 -> 82;

223 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
81 -> 223 [arrowhead = none ];
223 -> 83;

224 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 224 [arrowhead = none ];
224 -> 84;

225 [label="INSTANCE with matching:\nT4 -> T53\nT3 -> s(s(s(s(s(s(T54))))))\nT6 -> T56", fontsize=14, style = filled, fillcolor = lightgrey];
83 -> 225 [arrowhead = none , style=dashed];
225 -> 3[style=dashed];

226 [label="EVAL with clause\nhigher_valued(X100, X101) :- greater(s(X100), X101).\nand substitution\nT54 -> T57\nX100 -> s(s(s(s(s(s(T57))))))\nT52 -> T58\nX101 -> s(s(s(s(s(T58)))))", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 226 [arrowhead = none ];
226 -> 85;

227 [label="INSTANCE with matching:\nT37 -> s(s(s(s(s(s(s(T57)))))))\nT38 -> s(s(s(s(s(T58)))))", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 227 [arrowhead = none , style=dashed];
227 -> 64[style=dashed];

228 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
86 -> 228 [arrowhead = none ];
228 -> 87;

229 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
86 -> 229 [arrowhead = none ];
229 -> 88;

230 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 230 [arrowhead = none ];
230 -> 89;

231 [label="INSTANCE with matching:\nT4 -> T60\nT3 -> s(s(s(s(s(T61)))))\nT6 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
88 -> 231 [arrowhead = none , style=dashed];
231 -> 3[style=dashed];

232 [label="EVAL with clause\nhigher_valued(X112, X113) :- greater(s(X112), X113).\nand substitution\nT61 -> T64\nX112 -> s(s(s(s(s(T64)))))\nT59 -> T65\nX113 -> s(s(s(s(T65))))", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 232 [arrowhead = none ];
232 -> 90;

233 [label="INSTANCE with matching:\nT37 -> s(s(s(s(s(s(T64))))))\nT38 -> s(s(s(s(T65))))", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 233 [arrowhead = none , style=dashed];
233 -> 64[style=dashed];

234 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
91 -> 234 [arrowhead = none ];
234 -> 92;

235 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
91 -> 235 [arrowhead = none ];
235 -> 93;

236 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
92 -> 236 [arrowhead = none ];
236 -> 94;

237 [label="INSTANCE with matching:\nT4 -> T67\nT3 -> s(s(s(s(T68))))\nT6 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 237 [arrowhead = none , style=dashed];
237 -> 3[style=dashed];

238 [label="EVAL with clause\nhigher_valued(X124, X125) :- greater(s(X124), X125).\nand substitution\nT68 -> T71\nX124 -> s(s(s(s(T71))))\nT66 -> T72\nX125 -> s(s(s(T72)))", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 238 [arrowhead = none ];
238 -> 95;

239 [label="INSTANCE with matching:\nT37 -> s(s(s(s(s(T71)))))\nT38 -> s(s(s(T72)))", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 239 [arrowhead = none , style=dashed];
239 -> 64[style=dashed];

240 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
96 -> 240 [arrowhead = none ];
240 -> 97;

241 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
96 -> 241 [arrowhead = none ];
241 -> 98;

242 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
97 -> 242 [arrowhead = none ];
242 -> 99;

243 [label="INSTANCE with matching:\nT4 -> T74\nT3 -> s(s(s(T75)))\nT6 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 243 [arrowhead = none , style=dashed];
243 -> 3[style=dashed];

244 [label="EVAL with clause\nhigher_valued(X136, X137) :- greater(s(X136), X137).\nand substitution\nT75 -> T78\nX136 -> s(s(s(T78)))\nT73 -> T79\nX137 -> s(s(T79))", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 244 [arrowhead = none ];
244 -> 100;

245 [label="INSTANCE with matching:\nT37 -> s(s(s(s(T78))))\nT38 -> s(s(T79))", fontsize=14, style = filled, fillcolor = lightgrey];
100 -> 245 [arrowhead = none , style=dashed];
245 -> 64[style=dashed];

246 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
101 -> 246 [arrowhead = none ];
246 -> 102;

247 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
101 -> 247 [arrowhead = none ];
247 -> 103;

248 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 248 [arrowhead = none ];
248 -> 104;

249 [label="INSTANCE with matching:\nT4 -> T81\nT3 -> s(s(T82))\nT6 -> T84", fontsize=14, style = filled, fillcolor = lightgrey];
103 -> 249 [arrowhead = none , style=dashed];
249 -> 3[style=dashed];

250 [label="EVAL with clause\nhigher_valued(X148, X149) :- greater(s(X148), X149).\nand substitution\nT82 -> T85\nX148 -> s(s(T85))\nT80 -> T86\nX149 -> s(T86)", fontsize=14, style = filled, fillcolor = lightblue];
104 -> 250 [arrowhead = none ];
250 -> 105;

251 [label="INSTANCE with matching:\nT37 -> s(s(s(T85)))\nT38 -> s(T86)", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 251 [arrowhead = none , style=dashed];
251 -> 64[style=dashed];

252 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
106 -> 252 [arrowhead = none ];
252 -> 107;

253 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
106 -> 253 [arrowhead = none ];
253 -> 108;

254 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
107 -> 254 [arrowhead = none ];
254 -> 109;

255 [label="INSTANCE with matching:\nT4 -> T88\nT3 -> s(T89)\nT6 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
108 -> 255 [arrowhead = none , style=dashed];
255 -> 3[style=dashed];

256 [label="EVAL with clause\nhigher_valued(X160, X161) :- greater(s(X160), X161).\nand substitution\nT89 -> T92\nX160 -> s(T92)\nT87 -> T93\nX161 -> T93", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 256 [arrowhead = none ];
256 -> 110;

257 [label="INSTANCE with matching:\nT37 -> s(s(T92))\nT38 -> T93", fontsize=14, style = filled, fillcolor = lightgrey];
110 -> 257 [arrowhead = none , style=dashed];
257 -> 64[style=dashed];

258 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
111 -> 258 [arrowhead = none ];
258 -> 112;

259 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
111 -> 259 [arrowhead = none ];
259 -> 113;

260 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
112 -> 260 [arrowhead = none ];
260 -> 114;

261 [label="INSTANCE with matching:\nT4 -> T95\nT3 -> T96\nT6 -> T98", fontsize=14, style = filled, fillcolor = lightgrey];
113 -> 261 [arrowhead = none , style=dashed];
261 -> 3[style=dashed];

262 [label="EVAL with clause\nhigher_valued(X172, X173) :- greater(s(X172), X173).\nand substitution\nT96 -> T99\nX172 -> T99\nT94 -> T100\nX173 -> T100", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 262 [arrowhead = none ];
262 -> 115;

263 [label="INSTANCE with matching:\nT37 -> s(T99)\nT38 -> T100", fontsize=14, style = filled, fillcolor = lightgrey];
115 -> 263 [arrowhead = none , style=dashed];
263 -> 64[style=dashed];

264 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
116 -> 264 [arrowhead = none ];
264 -> 118;

265 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
116 -> 265 [arrowhead = none ];
265 -> 119;

266 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
118 -> 266 [arrowhead = none ];
266 -> 120;

267 [label="INSTANCE with matching:\nT4 -> T102\nT3 -> T103\nT6 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
119 -> 267 [arrowhead = none , style=dashed];
267 -> 3[style=dashed];

268 [label="EVAL with clause\nhigher_valued(X184, X185) :- greater(s(X184), X185).\nand substitution\nT103 -> T106\nX184 -> T106\nT101 -> T107\nX185 -> T107", fontsize=14, style = filled, fillcolor = lightblue];
120 -> 268 [arrowhead = none ];
268 -> 121;

269 [label="INSTANCE with matching:\nT37 -> s(T106)\nT38 -> T107", fontsize=14, style = filled, fillcolor = lightgrey];
121 -> 269 [arrowhead = none , style=dashed];
269 -> 64[style=dashed];

}

(2) Obligation:

Clauses:

max_valued3([], T7, T7).
max_valued3(.(T15, T9), 0, T12) :- max_valued3(T9, T15, T12).
max_valued3(.(s(T18), T9), s(0), T12) :- max_valued3(T9, s(T18), T12).
max_valued3(.(s(s(T21)), T9), s(s(0)), T12) :- max_valued3(T9, s(s(T21)), T12).
max_valued3(.(s(s(s(T24))), T9), s(s(s(0))), T12) :- max_valued3(T9, s(s(s(T24))), T12).
max_valued3(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) :- max_valued3(T9, s(s(s(s(T27)))), T12).
max_valued3(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) :- max_valued3(T9, s(s(s(s(s(T30))))), T12).
max_valued3(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) :- max_valued3(T9, s(s(s(s(s(s(T33)))))), T12).
max_valued3(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) :- max_valued3(T9, s(s(s(s(s(s(s(T36))))))), T12).
max_valued3(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) :- greater64(T37, T38).
max_valued3(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) :- ','(greater64(T37, T38), max_valued3(T9, s(T39), T12)).
max_valued3(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) :- higher_valued77(T48, T43).
max_valued3(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) :- ','(higher_valued77(T48, T43), max_valued3(T44, s(T49), T47)).
max_valued3(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) :- higher_valued82(T54, T52).
max_valued3(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) :- ','(higher_valued82(T54, T52), max_valued3(T53, s(s(s(s(s(s(T54)))))), T56)).
max_valued3(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) :- higher_valued87(T61, T59).
max_valued3(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) :- ','(higher_valued87(T61, T59), max_valued3(T60, s(s(s(s(s(T61))))), T63)).
max_valued3(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) :- higher_valued92(T68, T66).
max_valued3(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) :- ','(higher_valued92(T68, T66), max_valued3(T67, s(s(s(s(T68)))), T70)).
max_valued3(.(s(s(T73)), T74), s(s(s(T75))), T77) :- higher_valued97(T75, T73).
max_valued3(.(s(s(T73)), T74), s(s(s(T75))), T77) :- ','(higher_valued97(T75, T73), max_valued3(T74, s(s(s(T75))), T77)).
max_valued3(.(s(T80), T81), s(s(T82)), T84) :- higher_valued102(T82, T80).
max_valued3(.(s(T80), T81), s(s(T82)), T84) :- ','(higher_valued102(T82, T80), max_valued3(T81, s(s(T82)), T84)).
max_valued3(.(T87, T88), s(T89), T91) :- higher_valued107(T89, T87).
max_valued3(.(T87, T88), s(T89), T91) :- ','(higher_valued107(T89, T87), max_valued3(T88, s(T89), T91)).
max_valued3(.(T94, T95), T96, T98) :- higher_valued112(T96, T94).
max_valued3(.(T94, T95), T96, T98) :- ','(higher_valued112(T96, T94), max_valued3(T95, T96, T98)).
max_valued3(.(T101, T102), T103, T105) :- higher_valued118(T103, T101).
max_valued3(.(T101, T102), T103, T105) :- ','(higher_valued118(T103, T101), max_valued3(T102, T103, T105)).
greater64(s(T40), 0).
greater64(s(T41), s(T42)) :- greater64(T41, T42).
higher_valued77(T50, T51) :- greater64(s(s(T50)), s(s(s(s(s(s(T51))))))).
higher_valued82(T57, T58) :- greater64(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))).
higher_valued87(T64, T65) :- greater64(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))).
higher_valued92(T71, T72) :- greater64(s(s(s(s(s(T71))))), s(s(s(T72)))).
higher_valued97(T78, T79) :- greater64(s(s(s(s(T78)))), s(s(T79))).
higher_valued102(T85, T86) :- greater64(s(s(s(T85))), s(T86)).
higher_valued107(T92, T93) :- greater64(s(s(T92)), T93).
higher_valued112(T99, T100) :- greater64(s(T99), T100).
higher_valued118(T106, T107) :- greater64(s(T106), T107).
max_valued1(.(T7, []), T7).
max_valued1(.(0, .(T15, T9)), T12) :- max_valued3(T9, T15, T12).
max_valued1(.(s(0), .(s(T18), T9)), T12) :- max_valued3(T9, s(T18), T12).
max_valued1(.(s(s(0)), .(s(s(T21)), T9)), T12) :- max_valued3(T9, s(s(T21)), T12).
max_valued1(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) :- max_valued3(T9, s(s(s(T24))), T12).
max_valued1(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) :- max_valued3(T9, s(s(s(s(T27)))), T12).
max_valued1(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) :- max_valued3(T9, s(s(s(s(s(T30))))), T12).
max_valued1(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) :- max_valued3(T9, s(s(s(s(s(s(T33)))))), T12).
max_valued1(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) :- max_valued3(T9, s(s(s(s(s(s(s(T36))))))), T12).
max_valued1(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) :- greater64(T37, T38).
max_valued1(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) :- ','(greater64(T37, T38), max_valued3(T9, s(T39), T12)).
max_valued1(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) :- higher_valued77(T48, T43).
max_valued1(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) :- ','(higher_valued77(T48, T43), max_valued3(T44, s(T49), T47)).
max_valued1(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) :- higher_valued82(T54, T52).
max_valued1(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) :- ','(higher_valued82(T54, T52), max_valued3(T53, s(s(s(s(s(s(T54)))))), T56)).
max_valued1(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) :- higher_valued87(T61, T59).
max_valued1(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) :- ','(higher_valued87(T61, T59), max_valued3(T60, s(s(s(s(s(T61))))), T63)).
max_valued1(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) :- higher_valued92(T68, T66).
max_valued1(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) :- ','(higher_valued92(T68, T66), max_valued3(T67, s(s(s(s(T68)))), T70)).
max_valued1(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) :- higher_valued97(T75, T73).
max_valued1(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) :- ','(higher_valued97(T75, T73), max_valued3(T74, s(s(s(T75))), T77)).
max_valued1(.(s(s(T82)), .(s(T80), T81)), T84) :- higher_valued102(T82, T80).
max_valued1(.(s(s(T82)), .(s(T80), T81)), T84) :- ','(higher_valued102(T82, T80), max_valued3(T81, s(s(T82)), T84)).
max_valued1(.(s(T89), .(T87, T88)), T91) :- higher_valued107(T89, T87).
max_valued1(.(s(T89), .(T87, T88)), T91) :- ','(higher_valued107(T89, T87), max_valued3(T88, s(T89), T91)).
max_valued1(.(T96, .(T94, T95)), T98) :- higher_valued112(T96, T94).
max_valued1(.(T96, .(T94, T95)), T98) :- ','(higher_valued112(T96, T94), max_valued3(T95, T96, T98)).
max_valued1(.(T103, .(T101, T102)), T105) :- higher_valued118(T103, T101).
max_valued1(.(T103, .(T101, T102)), T105) :- ','(higher_valued118(T103, T101), max_valued3(T102, T103, T105)).

Queries:

max_valued1(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:
max_valued1_in: (b,f)
max_valued3_in: (b,b,f) (b,f,f)
greater64_in: (b,b) (f,b) (b,f)
higher_valued77_in: (f,b)
higher_valued82_in: (b,b) (f,b)
higher_valued87_in: (b,b) (f,b)
higher_valued92_in: (b,b) (f,b)
higher_valued97_in: (b,b) (f,b)
higher_valued102_in: (b,b) (f,b)
higher_valued107_in: (b,b) (f,b)
higher_valued112_in: (b,b) (f,b)
higher_valued118_in: (b,b) (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)

(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:

MAX_VALUED1_IN_GA(.(0, .(T15, T9)), T12) → U39_GA(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
MAX_VALUED1_IN_GA(.(0, .(T15, T9)), T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(T15, T9), 0, T12) → U1_GGA(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
MAX_VALUED3_IN_GGA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0), T12) → U2_GGA(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0)), T12) → U3_GGA(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_GGA(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_GGA(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_GGA(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_GGA(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_GGA(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GGA(T37, T9, T38, T12, greater64_in_gg(T37, T38))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → GREATER64_IN_GG(T37, T38)
GREATER64_IN_GG(s(T41), s(T42)) → U29_GG(T41, T42, greater64_in_gg(T41, T42))
GREATER64_IN_GG(s(T41), s(T42)) → GREATER64_IN_GG(T41, T42)
U9_GGA(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_GGA(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
U9_GGA(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(T15, T9), 0, T12) → U1_GAA(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
MAX_VALUED3_IN_GAA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GGA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → HIGHER_VALUED77_IN_AG(T48, T43)
HIGHER_VALUED77_IN_AG(T50, T51) → U30_AG(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
HIGHER_VALUED77_IN_AG(T50, T51) → GREATER64_IN_AG(s(s(T50)), s(s(s(s(s(s(T51)))))))
GREATER64_IN_AG(s(T41), s(T42)) → U29_AG(T41, T42, greater64_in_ag(T41, T42))
GREATER64_IN_AG(s(T41), s(T42)) → GREATER64_IN_AG(T41, T42)
U11_GGA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_GGA(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
U11_GGA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(T18), T9), s(0), T12) → U2_GAA(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
MAX_VALUED3_IN_GAA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GGA(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → HIGHER_VALUED82_IN_GG(T54, T52)
HIGHER_VALUED82_IN_GG(T57, T58) → U31_GG(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
HIGHER_VALUED82_IN_GG(T57, T58) → GREATER64_IN_GG(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))
U13_GGA(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_GGA(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U13_GGA(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GGA(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → HIGHER_VALUED87_IN_GG(T61, T59)
HIGHER_VALUED87_IN_GG(T64, T65) → U32_GG(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
HIGHER_VALUED87_IN_GG(T64, T65) → GREATER64_IN_GG(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))
U15_GGA(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_GGA(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U15_GGA(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GGA(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → HIGHER_VALUED92_IN_GG(T68, T66)
HIGHER_VALUED92_IN_GG(T71, T72) → U33_GG(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
HIGHER_VALUED92_IN_GG(T71, T72) → GREATER64_IN_GG(s(s(s(s(s(T71))))), s(s(s(T72))))
U17_GGA(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_GGA(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U17_GGA(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GGA(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75))), T77) → HIGHER_VALUED97_IN_GG(T75, T73)
HIGHER_VALUED97_IN_GG(T78, T79) → U34_GG(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
HIGHER_VALUED97_IN_GG(T78, T79) → GREATER64_IN_GG(s(s(s(s(T78)))), s(s(T79)))
U19_GGA(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_GGA(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U19_GGA(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82)), T84) → U21_GGA(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82)), T84) → HIGHER_VALUED102_IN_GG(T82, T80)
HIGHER_VALUED102_IN_GG(T85, T86) → U35_GG(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
HIGHER_VALUED102_IN_GG(T85, T86) → GREATER64_IN_GG(s(s(s(T85))), s(T86))
U21_GGA(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_GGA(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U21_GGA(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → MAX_VALUED3_IN_GGA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89), T91) → U23_GGA(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89), T91) → HIGHER_VALUED107_IN_GG(T89, T87)
HIGHER_VALUED107_IN_GG(T92, T93) → U36_GG(T92, T93, greater64_in_gg(s(s(T92)), T93))
HIGHER_VALUED107_IN_GG(T92, T93) → GREATER64_IN_GG(s(s(T92)), T93)
U23_GGA(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_GGA(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
U23_GGA(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → MAX_VALUED3_IN_GGA(T88, s(T89), T91)
MAX_VALUED3_IN_GGA(.(T94, T95), T96, T98) → U25_GGA(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
MAX_VALUED3_IN_GGA(.(T94, T95), T96, T98) → HIGHER_VALUED112_IN_GG(T96, T94)
HIGHER_VALUED112_IN_GG(T99, T100) → U37_GG(T99, T100, greater64_in_gg(s(T99), T100))
HIGHER_VALUED112_IN_GG(T99, T100) → GREATER64_IN_GG(s(T99), T100)
U25_GGA(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_GGA(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
U25_GGA(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → MAX_VALUED3_IN_GGA(T95, T96, T98)
MAX_VALUED3_IN_GGA(.(T101, T102), T103, T105) → U27_GGA(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
MAX_VALUED3_IN_GGA(.(T101, T102), T103, T105) → HIGHER_VALUED118_IN_GG(T103, T101)
HIGHER_VALUED118_IN_GG(T106, T107) → U38_GG(T106, T107, greater64_in_gg(s(T106), T107))
HIGHER_VALUED118_IN_GG(T106, T107) → GREATER64_IN_GG(s(T106), T107)
U27_GGA(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_GGA(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U27_GGA(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → MAX_VALUED3_IN_GGA(T102, T103, T105)
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9), s(s(0)), T12) → U3_GAA(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_GAA(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_GAA(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_GAA(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_GAA(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_GAA(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GAA(T37, T9, T38, T12, greater64_in_ga(T37, T38))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → GREATER64_IN_GA(T37, T38)
GREATER64_IN_GA(s(T41), s(T42)) → U29_GA(T41, T42, greater64_in_ga(T41, T42))
GREATER64_IN_GA(s(T41), s(T42)) → GREATER64_IN_GA(T41, T42)
U9_GAA(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_GAA(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
U9_GAA(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GAA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → HIGHER_VALUED77_IN_AG(T48, T43)
U11_GAA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_GAA(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
U11_GAA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GAA(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → HIGHER_VALUED82_IN_AG(T54, T52)
HIGHER_VALUED82_IN_AG(T57, T58) → U31_AG(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
HIGHER_VALUED82_IN_AG(T57, T58) → GREATER64_IN_AG(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))
U13_GAA(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_GAA(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
U13_GAA(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → MAX_VALUED3_IN_GAA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GAA(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → HIGHER_VALUED87_IN_AG(T61, T59)
HIGHER_VALUED87_IN_AG(T64, T65) → U32_AG(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
HIGHER_VALUED87_IN_AG(T64, T65) → GREATER64_IN_AG(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))
U15_GAA(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_GAA(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
U15_GAA(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → MAX_VALUED3_IN_GAA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GAA(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → HIGHER_VALUED92_IN_AG(T68, T66)
HIGHER_VALUED92_IN_AG(T71, T72) → U33_AG(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
HIGHER_VALUED92_IN_AG(T71, T72) → GREATER64_IN_AG(s(s(s(s(s(T71))))), s(s(s(T72))))
U17_GAA(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_GAA(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
U17_GAA(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → MAX_VALUED3_IN_GAA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GAA(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74), s(s(s(T75))), T77) → HIGHER_VALUED97_IN_AG(T75, T73)
HIGHER_VALUED97_IN_AG(T78, T79) → U34_AG(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
HIGHER_VALUED97_IN_AG(T78, T79) → GREATER64_IN_AG(s(s(s(s(T78)))), s(s(T79)))
U19_GAA(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_GAA(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
U19_GAA(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → MAX_VALUED3_IN_GAA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GAA(.(s(T80), T81), s(s(T82)), T84) → U21_GAA(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
MAX_VALUED3_IN_GAA(.(s(T80), T81), s(s(T82)), T84) → HIGHER_VALUED102_IN_AG(T82, T80)
HIGHER_VALUED102_IN_AG(T85, T86) → U35_AG(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
HIGHER_VALUED102_IN_AG(T85, T86) → GREATER64_IN_AG(s(s(s(T85))), s(T86))
U21_GAA(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_GAA(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
U21_GAA(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → MAX_VALUED3_IN_GAA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GAA(.(T87, T88), s(T89), T91) → U23_GAA(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
MAX_VALUED3_IN_GAA(.(T87, T88), s(T89), T91) → HIGHER_VALUED107_IN_AG(T89, T87)
HIGHER_VALUED107_IN_AG(T92, T93) → U36_AG(T92, T93, greater64_in_ag(s(s(T92)), T93))
HIGHER_VALUED107_IN_AG(T92, T93) → GREATER64_IN_AG(s(s(T92)), T93)
U23_GAA(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_GAA(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
U23_GAA(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → MAX_VALUED3_IN_GAA(T88, s(T89), T91)
MAX_VALUED3_IN_GAA(.(T94, T95), T96, T98) → U25_GAA(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
MAX_VALUED3_IN_GAA(.(T94, T95), T96, T98) → HIGHER_VALUED112_IN_AG(T96, T94)
HIGHER_VALUED112_IN_AG(T99, T100) → U37_AG(T99, T100, greater64_in_ag(s(T99), T100))
HIGHER_VALUED112_IN_AG(T99, T100) → GREATER64_IN_AG(s(T99), T100)
U25_GAA(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_GAA(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
U25_GAA(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → MAX_VALUED3_IN_GAA(T95, T96, T98)
MAX_VALUED3_IN_GAA(.(T101, T102), T103, T105) → U27_GAA(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
MAX_VALUED3_IN_GAA(.(T101, T102), T103, T105) → HIGHER_VALUED118_IN_AG(T103, T101)
HIGHER_VALUED118_IN_AG(T106, T107) → U38_AG(T106, T107, greater64_in_ag(s(T106), T107))
HIGHER_VALUED118_IN_AG(T106, T107) → GREATER64_IN_AG(s(T106), T107)
U27_GAA(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_GAA(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U27_GAA(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → MAX_VALUED3_IN_GAA(T102, T103, T105)
MAX_VALUED1_IN_GA(.(s(0), .(s(T18), T9)), T12) → U40_GA(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
MAX_VALUED1_IN_GA(.(s(0), .(s(T18), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED1_IN_GA(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_GA(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
MAX_VALUED1_IN_GA(.(s(s(0)), .(s(s(T21)), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED1_IN_GA(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_GA(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_GA(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_GA(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_GA(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_GA(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_GA(T38, T37, T9, T12, greater64_in_gg(T37, T38))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → GREATER64_IN_GG(T37, T38)
U47_GA(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_GA(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U47_GA(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_GA(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → HIGHER_VALUED77_IN_AG(T48, T43)
U49_GA(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_GA(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U49_GA(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_GA(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → HIGHER_VALUED82_IN_GG(T54, T52)
U51_GA(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_GA(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U51_GA(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_GA(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → HIGHER_VALUED87_IN_GG(T61, T59)
U53_GA(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_GA(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U53_GA(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED1_IN_GA(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_GA(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
MAX_VALUED1_IN_GA(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → HIGHER_VALUED92_IN_GG(T68, T66)
U55_GA(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_GA(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U55_GA(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED1_IN_GA(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_GA(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
MAX_VALUED1_IN_GA(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → HIGHER_VALUED97_IN_GG(T75, T73)
U57_GA(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_GA(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U57_GA(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))), T77)
MAX_VALUED1_IN_GA(.(s(s(T82)), .(s(T80), T81)), T84) → U59_GA(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
MAX_VALUED1_IN_GA(.(s(s(T82)), .(s(T80), T81)), T84) → HIGHER_VALUED102_IN_GG(T82, T80)
U59_GA(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_GA(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U59_GA(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → MAX_VALUED3_IN_GGA(T81, s(s(T82)), T84)
MAX_VALUED1_IN_GA(.(s(T89), .(T87, T88)), T91) → U61_GA(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
MAX_VALUED1_IN_GA(.(s(T89), .(T87, T88)), T91) → HIGHER_VALUED107_IN_GG(T89, T87)
U61_GA(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_GA(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U61_GA(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → MAX_VALUED3_IN_GGA(T88, s(T89), T91)
MAX_VALUED1_IN_GA(.(T96, .(T94, T95)), T98) → U63_GA(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
MAX_VALUED1_IN_GA(.(T96, .(T94, T95)), T98) → HIGHER_VALUED112_IN_GG(T96, T94)
U63_GA(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_GA(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U63_GA(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → MAX_VALUED3_IN_GGA(T95, T96, T98)
MAX_VALUED1_IN_GA(.(T103, .(T101, T102)), T105) → U65_GA(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
MAX_VALUED1_IN_GA(.(T103, .(T101, T102)), T105) → HIGHER_VALUED118_IN_GG(T103, T101)
U65_GA(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_GA(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U65_GA(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → MAX_VALUED3_IN_GGA(T102, T103, T105)

The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)
MAX_VALUED1_IN_GA(x1, x2)  =  MAX_VALUED1_IN_GA(x1)
U39_GA(x1, x2, x3, x4)  =  U39_GA(x4)
MAX_VALUED3_IN_GGA(x1, x2, x3)  =  MAX_VALUED3_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x4)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x2, x5)
GREATER64_IN_GG(x1, x2)  =  GREATER64_IN_GG(x1, x2)
U29_GG(x1, x2, x3)  =  U29_GG(x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x5)
MAX_VALUED3_IN_GAA(x1, x2, x3)  =  MAX_VALUED3_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x2, x5)
HIGHER_VALUED77_IN_AG(x1, x2)  =  HIGHER_VALUED77_IN_AG(x2)
U30_AG(x1, x2, x3)  =  U30_AG(x3)
GREATER64_IN_AG(x1, x2)  =  GREATER64_IN_AG(x2)
U29_AG(x1, x2, x3)  =  U29_AG(x3)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x5)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x2, x3, x5)
HIGHER_VALUED82_IN_GG(x1, x2)  =  HIGHER_VALUED82_IN_GG(x1, x2)
U31_GG(x1, x2, x3)  =  U31_GG(x3)
U14_GGA(x1, x2, x3, x4, x5)  =  U14_GGA(x5)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x2, x3, x5)
HIGHER_VALUED87_IN_GG(x1, x2)  =  HIGHER_VALUED87_IN_GG(x1, x2)
U32_GG(x1, x2, x3)  =  U32_GG(x3)
U16_GGA(x1, x2, x3, x4, x5)  =  U16_GGA(x5)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x2, x3, x5)
HIGHER_VALUED92_IN_GG(x1, x2)  =  HIGHER_VALUED92_IN_GG(x1, x2)
U33_GG(x1, x2, x3)  =  U33_GG(x3)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x2, x3, x5)
HIGHER_VALUED97_IN_GG(x1, x2)  =  HIGHER_VALUED97_IN_GG(x1, x2)
U34_GG(x1, x2, x3)  =  U34_GG(x3)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x5)
U21_GGA(x1, x2, x3, x4, x5)  =  U21_GGA(x2, x3, x5)
HIGHER_VALUED102_IN_GG(x1, x2)  =  HIGHER_VALUED102_IN_GG(x1, x2)
U35_GG(x1, x2, x3)  =  U35_GG(x3)
U22_GGA(x1, x2, x3, x4, x5)  =  U22_GGA(x5)
U23_GGA(x1, x2, x3, x4, x5)  =  U23_GGA(x2, x3, x5)
HIGHER_VALUED107_IN_GG(x1, x2)  =  HIGHER_VALUED107_IN_GG(x1, x2)
U36_GG(x1, x2, x3)  =  U36_GG(x3)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(x5)
U25_GGA(x1, x2, x3, x4, x5)  =  U25_GGA(x2, x3, x5)
HIGHER_VALUED112_IN_GG(x1, x2)  =  HIGHER_VALUED112_IN_GG(x1, x2)
U37_GG(x1, x2, x3)  =  U37_GG(x3)
U26_GGA(x1, x2, x3, x4, x5)  =  U26_GGA(x5)
U27_GGA(x1, x2, x3, x4, x5)  =  U27_GGA(x2, x3, x5)
HIGHER_VALUED118_IN_GG(x1, x2)  =  HIGHER_VALUED118_IN_GG(x1, x2)
U38_GG(x1, x2, x3)  =  U38_GG(x3)
U28_GGA(x1, x2, x3, x4, x5)  =  U28_GGA(x5)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x2, x5)
GREATER64_IN_GA(x1, x2)  =  GREATER64_IN_GA(x1)
U29_GA(x1, x2, x3)  =  U29_GA(x3)
U10_GAA(x1, x2, x3, x4, x5)  =  U10_GAA(x5)
U11_GAA(x1, x2, x3, x4, x5)  =  U11_GAA(x2, x5)
U12_GAA(x1, x2, x3, x4, x5)  =  U12_GAA(x5)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x2, x5)
HIGHER_VALUED82_IN_AG(x1, x2)  =  HIGHER_VALUED82_IN_AG(x2)
U31_AG(x1, x2, x3)  =  U31_AG(x3)
U14_GAA(x1, x2, x3, x4, x5)  =  U14_GAA(x5)
U15_GAA(x1, x2, x3, x4, x5)  =  U15_GAA(x2, x5)
HIGHER_VALUED87_IN_AG(x1, x2)  =  HIGHER_VALUED87_IN_AG(x2)
U32_AG(x1, x2, x3)  =  U32_AG(x3)
U16_GAA(x1, x2, x3, x4, x5)  =  U16_GAA(x5)
U17_GAA(x1, x2, x3, x4, x5)  =  U17_GAA(x2, x5)
HIGHER_VALUED92_IN_AG(x1, x2)  =  HIGHER_VALUED92_IN_AG(x2)
U33_AG(x1, x2, x3)  =  U33_AG(x3)
U18_GAA(x1, x2, x3, x4, x5)  =  U18_GAA(x5)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x2, x5)
HIGHER_VALUED97_IN_AG(x1, x2)  =  HIGHER_VALUED97_IN_AG(x2)
U34_AG(x1, x2, x3)  =  U34_AG(x3)
U20_GAA(x1, x2, x3, x4, x5)  =  U20_GAA(x5)
U21_GAA(x1, x2, x3, x4, x5)  =  U21_GAA(x2, x5)
HIGHER_VALUED102_IN_AG(x1, x2)  =  HIGHER_VALUED102_IN_AG(x2)
U35_AG(x1, x2, x3)  =  U35_AG(x3)
U22_GAA(x1, x2, x3, x4, x5)  =  U22_GAA(x5)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x2, x5)
HIGHER_VALUED107_IN_AG(x1, x2)  =  HIGHER_VALUED107_IN_AG(x2)
U36_AG(x1, x2, x3)  =  U36_AG(x3)
U24_GAA(x1, x2, x3, x4, x5)  =  U24_GAA(x5)
U25_GAA(x1, x2, x3, x4, x5)  =  U25_GAA(x2, x5)
HIGHER_VALUED112_IN_AG(x1, x2)  =  HIGHER_VALUED112_IN_AG(x2)
U37_AG(x1, x2, x3)  =  U37_AG(x3)
U26_GAA(x1, x2, x3, x4, x5)  =  U26_GAA(x5)
U27_GAA(x1, x2, x3, x4, x5)  =  U27_GAA(x2, x5)
HIGHER_VALUED118_IN_AG(x1, x2)  =  HIGHER_VALUED118_IN_AG(x2)
U38_AG(x1, x2, x3)  =  U38_AG(x3)
U28_GAA(x1, x2, x3, x4, x5)  =  U28_GAA(x5)
U40_GA(x1, x2, x3, x4)  =  U40_GA(x4)
U41_GA(x1, x2, x3, x4)  =  U41_GA(x4)
U42_GA(x1, x2, x3, x4)  =  U42_GA(x4)
U43_GA(x1, x2, x3, x4)  =  U43_GA(x4)
U44_GA(x1, x2, x3, x4)  =  U44_GA(x4)
U45_GA(x1, x2, x3, x4)  =  U45_GA(x4)
U46_GA(x1, x2, x3, x4)  =  U46_GA(x4)
U47_GA(x1, x2, x3, x4, x5)  =  U47_GA(x3, x5)
U48_GA(x1, x2, x3, x4, x5)  =  U48_GA(x5)
U49_GA(x1, x2, x3, x4, x5)  =  U49_GA(x3, x5)
U50_GA(x1, x2, x3, x4, x5)  =  U50_GA(x5)
U51_GA(x1, x2, x3, x4, x5)  =  U51_GA(x1, x3, x5)
U52_GA(x1, x2, x3, x4, x5)  =  U52_GA(x5)
U53_GA(x1, x2, x3, x4, x5)  =  U53_GA(x1, x3, x5)
U54_GA(x1, x2, x3, x4, x5)  =  U54_GA(x5)
U55_GA(x1, x2, x3, x4, x5)  =  U55_GA(x1, x3, x5)
U56_GA(x1, x2, x3, x4, x5)  =  U56_GA(x5)
U57_GA(x1, x2, x3, x4, x5)  =  U57_GA(x1, x3, x5)
U58_GA(x1, x2, x3, x4, x5)  =  U58_GA(x5)
U59_GA(x1, x2, x3, x4, x5)  =  U59_GA(x1, x3, x5)
U60_GA(x1, x2, x3, x4, x5)  =  U60_GA(x5)
U61_GA(x1, x2, x3, x4, x5)  =  U61_GA(x1, x3, x5)
U62_GA(x1, x2, x3, x4, x5)  =  U62_GA(x5)
U63_GA(x1, x2, x3, x4, x5)  =  U63_GA(x1, x3, x5)
U64_GA(x1, x2, x3, x4, x5)  =  U64_GA(x5)
U65_GA(x1, x2, x3, x4, x5)  =  U65_GA(x1, x3, x5)
U66_GA(x1, x2, x3, x4, x5)  =  U66_GA(x5)

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

(6) Obligation:

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

MAX_VALUED1_IN_GA(.(0, .(T15, T9)), T12) → U39_GA(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
MAX_VALUED1_IN_GA(.(0, .(T15, T9)), T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(T15, T9), 0, T12) → U1_GGA(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
MAX_VALUED3_IN_GGA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0), T12) → U2_GGA(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0)), T12) → U3_GGA(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_GGA(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_GGA(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_GGA(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_GGA(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_GGA(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GGA(T37, T9, T38, T12, greater64_in_gg(T37, T38))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → GREATER64_IN_GG(T37, T38)
GREATER64_IN_GG(s(T41), s(T42)) → U29_GG(T41, T42, greater64_in_gg(T41, T42))
GREATER64_IN_GG(s(T41), s(T42)) → GREATER64_IN_GG(T41, T42)
U9_GGA(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_GGA(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
U9_GGA(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(T15, T9), 0, T12) → U1_GAA(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
MAX_VALUED3_IN_GAA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GGA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → HIGHER_VALUED77_IN_AG(T48, T43)
HIGHER_VALUED77_IN_AG(T50, T51) → U30_AG(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
HIGHER_VALUED77_IN_AG(T50, T51) → GREATER64_IN_AG(s(s(T50)), s(s(s(s(s(s(T51)))))))
GREATER64_IN_AG(s(T41), s(T42)) → U29_AG(T41, T42, greater64_in_ag(T41, T42))
GREATER64_IN_AG(s(T41), s(T42)) → GREATER64_IN_AG(T41, T42)
U11_GGA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_GGA(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
U11_GGA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(T18), T9), s(0), T12) → U2_GAA(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
MAX_VALUED3_IN_GAA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GGA(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → HIGHER_VALUED82_IN_GG(T54, T52)
HIGHER_VALUED82_IN_GG(T57, T58) → U31_GG(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
HIGHER_VALUED82_IN_GG(T57, T58) → GREATER64_IN_GG(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))
U13_GGA(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_GGA(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U13_GGA(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GGA(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → HIGHER_VALUED87_IN_GG(T61, T59)
HIGHER_VALUED87_IN_GG(T64, T65) → U32_GG(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
HIGHER_VALUED87_IN_GG(T64, T65) → GREATER64_IN_GG(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))
U15_GGA(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_GGA(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U15_GGA(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GGA(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → HIGHER_VALUED92_IN_GG(T68, T66)
HIGHER_VALUED92_IN_GG(T71, T72) → U33_GG(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
HIGHER_VALUED92_IN_GG(T71, T72) → GREATER64_IN_GG(s(s(s(s(s(T71))))), s(s(s(T72))))
U17_GGA(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_GGA(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U17_GGA(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GGA(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75))), T77) → HIGHER_VALUED97_IN_GG(T75, T73)
HIGHER_VALUED97_IN_GG(T78, T79) → U34_GG(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
HIGHER_VALUED97_IN_GG(T78, T79) → GREATER64_IN_GG(s(s(s(s(T78)))), s(s(T79)))
U19_GGA(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_GGA(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U19_GGA(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82)), T84) → U21_GGA(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82)), T84) → HIGHER_VALUED102_IN_GG(T82, T80)
HIGHER_VALUED102_IN_GG(T85, T86) → U35_GG(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
HIGHER_VALUED102_IN_GG(T85, T86) → GREATER64_IN_GG(s(s(s(T85))), s(T86))
U21_GGA(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_GGA(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U21_GGA(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → MAX_VALUED3_IN_GGA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89), T91) → U23_GGA(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89), T91) → HIGHER_VALUED107_IN_GG(T89, T87)
HIGHER_VALUED107_IN_GG(T92, T93) → U36_GG(T92, T93, greater64_in_gg(s(s(T92)), T93))
HIGHER_VALUED107_IN_GG(T92, T93) → GREATER64_IN_GG(s(s(T92)), T93)
U23_GGA(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_GGA(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
U23_GGA(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → MAX_VALUED3_IN_GGA(T88, s(T89), T91)
MAX_VALUED3_IN_GGA(.(T94, T95), T96, T98) → U25_GGA(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
MAX_VALUED3_IN_GGA(.(T94, T95), T96, T98) → HIGHER_VALUED112_IN_GG(T96, T94)
HIGHER_VALUED112_IN_GG(T99, T100) → U37_GG(T99, T100, greater64_in_gg(s(T99), T100))
HIGHER_VALUED112_IN_GG(T99, T100) → GREATER64_IN_GG(s(T99), T100)
U25_GGA(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_GGA(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
U25_GGA(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → MAX_VALUED3_IN_GGA(T95, T96, T98)
MAX_VALUED3_IN_GGA(.(T101, T102), T103, T105) → U27_GGA(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
MAX_VALUED3_IN_GGA(.(T101, T102), T103, T105) → HIGHER_VALUED118_IN_GG(T103, T101)
HIGHER_VALUED118_IN_GG(T106, T107) → U38_GG(T106, T107, greater64_in_gg(s(T106), T107))
HIGHER_VALUED118_IN_GG(T106, T107) → GREATER64_IN_GG(s(T106), T107)
U27_GGA(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_GGA(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U27_GGA(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → MAX_VALUED3_IN_GGA(T102, T103, T105)
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9), s(s(0)), T12) → U3_GAA(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_GAA(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_GAA(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_GAA(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_GAA(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_GAA(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GAA(T37, T9, T38, T12, greater64_in_ga(T37, T38))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → GREATER64_IN_GA(T37, T38)
GREATER64_IN_GA(s(T41), s(T42)) → U29_GA(T41, T42, greater64_in_ga(T41, T42))
GREATER64_IN_GA(s(T41), s(T42)) → GREATER64_IN_GA(T41, T42)
U9_GAA(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_GAA(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
U9_GAA(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GAA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → HIGHER_VALUED77_IN_AG(T48, T43)
U11_GAA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_GAA(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
U11_GAA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GAA(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → HIGHER_VALUED82_IN_AG(T54, T52)
HIGHER_VALUED82_IN_AG(T57, T58) → U31_AG(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
HIGHER_VALUED82_IN_AG(T57, T58) → GREATER64_IN_AG(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))
U13_GAA(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_GAA(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
U13_GAA(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → MAX_VALUED3_IN_GAA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GAA(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → HIGHER_VALUED87_IN_AG(T61, T59)
HIGHER_VALUED87_IN_AG(T64, T65) → U32_AG(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
HIGHER_VALUED87_IN_AG(T64, T65) → GREATER64_IN_AG(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))
U15_GAA(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_GAA(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
U15_GAA(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → MAX_VALUED3_IN_GAA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GAA(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → HIGHER_VALUED92_IN_AG(T68, T66)
HIGHER_VALUED92_IN_AG(T71, T72) → U33_AG(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
HIGHER_VALUED92_IN_AG(T71, T72) → GREATER64_IN_AG(s(s(s(s(s(T71))))), s(s(s(T72))))
U17_GAA(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_GAA(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
U17_GAA(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → MAX_VALUED3_IN_GAA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GAA(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74), s(s(s(T75))), T77) → HIGHER_VALUED97_IN_AG(T75, T73)
HIGHER_VALUED97_IN_AG(T78, T79) → U34_AG(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
HIGHER_VALUED97_IN_AG(T78, T79) → GREATER64_IN_AG(s(s(s(s(T78)))), s(s(T79)))
U19_GAA(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_GAA(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
U19_GAA(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → MAX_VALUED3_IN_GAA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GAA(.(s(T80), T81), s(s(T82)), T84) → U21_GAA(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
MAX_VALUED3_IN_GAA(.(s(T80), T81), s(s(T82)), T84) → HIGHER_VALUED102_IN_AG(T82, T80)
HIGHER_VALUED102_IN_AG(T85, T86) → U35_AG(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
HIGHER_VALUED102_IN_AG(T85, T86) → GREATER64_IN_AG(s(s(s(T85))), s(T86))
U21_GAA(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_GAA(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
U21_GAA(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → MAX_VALUED3_IN_GAA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GAA(.(T87, T88), s(T89), T91) → U23_GAA(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
MAX_VALUED3_IN_GAA(.(T87, T88), s(T89), T91) → HIGHER_VALUED107_IN_AG(T89, T87)
HIGHER_VALUED107_IN_AG(T92, T93) → U36_AG(T92, T93, greater64_in_ag(s(s(T92)), T93))
HIGHER_VALUED107_IN_AG(T92, T93) → GREATER64_IN_AG(s(s(T92)), T93)
U23_GAA(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_GAA(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
U23_GAA(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → MAX_VALUED3_IN_GAA(T88, s(T89), T91)
MAX_VALUED3_IN_GAA(.(T94, T95), T96, T98) → U25_GAA(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
MAX_VALUED3_IN_GAA(.(T94, T95), T96, T98) → HIGHER_VALUED112_IN_AG(T96, T94)
HIGHER_VALUED112_IN_AG(T99, T100) → U37_AG(T99, T100, greater64_in_ag(s(T99), T100))
HIGHER_VALUED112_IN_AG(T99, T100) → GREATER64_IN_AG(s(T99), T100)
U25_GAA(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_GAA(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
U25_GAA(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → MAX_VALUED3_IN_GAA(T95, T96, T98)
MAX_VALUED3_IN_GAA(.(T101, T102), T103, T105) → U27_GAA(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
MAX_VALUED3_IN_GAA(.(T101, T102), T103, T105) → HIGHER_VALUED118_IN_AG(T103, T101)
HIGHER_VALUED118_IN_AG(T106, T107) → U38_AG(T106, T107, greater64_in_ag(s(T106), T107))
HIGHER_VALUED118_IN_AG(T106, T107) → GREATER64_IN_AG(s(T106), T107)
U27_GAA(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_GAA(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U27_GAA(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → MAX_VALUED3_IN_GAA(T102, T103, T105)
MAX_VALUED1_IN_GA(.(s(0), .(s(T18), T9)), T12) → U40_GA(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
MAX_VALUED1_IN_GA(.(s(0), .(s(T18), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED1_IN_GA(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_GA(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
MAX_VALUED1_IN_GA(.(s(s(0)), .(s(s(T21)), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED1_IN_GA(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_GA(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_GA(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_GA(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_GA(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_GA(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_GA(T38, T37, T9, T12, greater64_in_gg(T37, T38))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → GREATER64_IN_GG(T37, T38)
U47_GA(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_GA(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U47_GA(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_GA(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → HIGHER_VALUED77_IN_AG(T48, T43)
U49_GA(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_GA(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U49_GA(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_GA(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → HIGHER_VALUED82_IN_GG(T54, T52)
U51_GA(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_GA(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U51_GA(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED1_IN_GA(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_GA(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
MAX_VALUED1_IN_GA(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → HIGHER_VALUED87_IN_GG(T61, T59)
U53_GA(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_GA(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U53_GA(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED1_IN_GA(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_GA(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
MAX_VALUED1_IN_GA(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → HIGHER_VALUED92_IN_GG(T68, T66)
U55_GA(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_GA(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U55_GA(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED1_IN_GA(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_GA(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
MAX_VALUED1_IN_GA(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → HIGHER_VALUED97_IN_GG(T75, T73)
U57_GA(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_GA(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U57_GA(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))), T77)
MAX_VALUED1_IN_GA(.(s(s(T82)), .(s(T80), T81)), T84) → U59_GA(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
MAX_VALUED1_IN_GA(.(s(s(T82)), .(s(T80), T81)), T84) → HIGHER_VALUED102_IN_GG(T82, T80)
U59_GA(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_GA(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U59_GA(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → MAX_VALUED3_IN_GGA(T81, s(s(T82)), T84)
MAX_VALUED1_IN_GA(.(s(T89), .(T87, T88)), T91) → U61_GA(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
MAX_VALUED1_IN_GA(.(s(T89), .(T87, T88)), T91) → HIGHER_VALUED107_IN_GG(T89, T87)
U61_GA(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_GA(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U61_GA(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → MAX_VALUED3_IN_GGA(T88, s(T89), T91)
MAX_VALUED1_IN_GA(.(T96, .(T94, T95)), T98) → U63_GA(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
MAX_VALUED1_IN_GA(.(T96, .(T94, T95)), T98) → HIGHER_VALUED112_IN_GG(T96, T94)
U63_GA(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_GA(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U63_GA(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → MAX_VALUED3_IN_GGA(T95, T96, T98)
MAX_VALUED1_IN_GA(.(T103, .(T101, T102)), T105) → U65_GA(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
MAX_VALUED1_IN_GA(.(T103, .(T101, T102)), T105) → HIGHER_VALUED118_IN_GG(T103, T101)
U65_GA(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_GA(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U65_GA(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → MAX_VALUED3_IN_GGA(T102, T103, T105)

The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)
MAX_VALUED1_IN_GA(x1, x2)  =  MAX_VALUED1_IN_GA(x1)
U39_GA(x1, x2, x3, x4)  =  U39_GA(x4)
MAX_VALUED3_IN_GGA(x1, x2, x3)  =  MAX_VALUED3_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x4)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x2, x5)
GREATER64_IN_GG(x1, x2)  =  GREATER64_IN_GG(x1, x2)
U29_GG(x1, x2, x3)  =  U29_GG(x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x5)
MAX_VALUED3_IN_GAA(x1, x2, x3)  =  MAX_VALUED3_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x2, x5)
HIGHER_VALUED77_IN_AG(x1, x2)  =  HIGHER_VALUED77_IN_AG(x2)
U30_AG(x1, x2, x3)  =  U30_AG(x3)
GREATER64_IN_AG(x1, x2)  =  GREATER64_IN_AG(x2)
U29_AG(x1, x2, x3)  =  U29_AG(x3)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x5)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x2, x3, x5)
HIGHER_VALUED82_IN_GG(x1, x2)  =  HIGHER_VALUED82_IN_GG(x1, x2)
U31_GG(x1, x2, x3)  =  U31_GG(x3)
U14_GGA(x1, x2, x3, x4, x5)  =  U14_GGA(x5)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x2, x3, x5)
HIGHER_VALUED87_IN_GG(x1, x2)  =  HIGHER_VALUED87_IN_GG(x1, x2)
U32_GG(x1, x2, x3)  =  U32_GG(x3)
U16_GGA(x1, x2, x3, x4, x5)  =  U16_GGA(x5)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x2, x3, x5)
HIGHER_VALUED92_IN_GG(x1, x2)  =  HIGHER_VALUED92_IN_GG(x1, x2)
U33_GG(x1, x2, x3)  =  U33_GG(x3)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x2, x3, x5)
HIGHER_VALUED97_IN_GG(x1, x2)  =  HIGHER_VALUED97_IN_GG(x1, x2)
U34_GG(x1, x2, x3)  =  U34_GG(x3)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x5)
U21_GGA(x1, x2, x3, x4, x5)  =  U21_GGA(x2, x3, x5)
HIGHER_VALUED102_IN_GG(x1, x2)  =  HIGHER_VALUED102_IN_GG(x1, x2)
U35_GG(x1, x2, x3)  =  U35_GG(x3)
U22_GGA(x1, x2, x3, x4, x5)  =  U22_GGA(x5)
U23_GGA(x1, x2, x3, x4, x5)  =  U23_GGA(x2, x3, x5)
HIGHER_VALUED107_IN_GG(x1, x2)  =  HIGHER_VALUED107_IN_GG(x1, x2)
U36_GG(x1, x2, x3)  =  U36_GG(x3)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(x5)
U25_GGA(x1, x2, x3, x4, x5)  =  U25_GGA(x2, x3, x5)
HIGHER_VALUED112_IN_GG(x1, x2)  =  HIGHER_VALUED112_IN_GG(x1, x2)
U37_GG(x1, x2, x3)  =  U37_GG(x3)
U26_GGA(x1, x2, x3, x4, x5)  =  U26_GGA(x5)
U27_GGA(x1, x2, x3, x4, x5)  =  U27_GGA(x2, x3, x5)
HIGHER_VALUED118_IN_GG(x1, x2)  =  HIGHER_VALUED118_IN_GG(x1, x2)
U38_GG(x1, x2, x3)  =  U38_GG(x3)
U28_GGA(x1, x2, x3, x4, x5)  =  U28_GGA(x5)
U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x4)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x4)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x2, x5)
GREATER64_IN_GA(x1, x2)  =  GREATER64_IN_GA(x1)
U29_GA(x1, x2, x3)  =  U29_GA(x3)
U10_GAA(x1, x2, x3, x4, x5)  =  U10_GAA(x5)
U11_GAA(x1, x2, x3, x4, x5)  =  U11_GAA(x2, x5)
U12_GAA(x1, x2, x3, x4, x5)  =  U12_GAA(x5)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x2, x5)
HIGHER_VALUED82_IN_AG(x1, x2)  =  HIGHER_VALUED82_IN_AG(x2)
U31_AG(x1, x2, x3)  =  U31_AG(x3)
U14_GAA(x1, x2, x3, x4, x5)  =  U14_GAA(x5)
U15_GAA(x1, x2, x3, x4, x5)  =  U15_GAA(x2, x5)
HIGHER_VALUED87_IN_AG(x1, x2)  =  HIGHER_VALUED87_IN_AG(x2)
U32_AG(x1, x2, x3)  =  U32_AG(x3)
U16_GAA(x1, x2, x3, x4, x5)  =  U16_GAA(x5)
U17_GAA(x1, x2, x3, x4, x5)  =  U17_GAA(x2, x5)
HIGHER_VALUED92_IN_AG(x1, x2)  =  HIGHER_VALUED92_IN_AG(x2)
U33_AG(x1, x2, x3)  =  U33_AG(x3)
U18_GAA(x1, x2, x3, x4, x5)  =  U18_GAA(x5)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x2, x5)
HIGHER_VALUED97_IN_AG(x1, x2)  =  HIGHER_VALUED97_IN_AG(x2)
U34_AG(x1, x2, x3)  =  U34_AG(x3)
U20_GAA(x1, x2, x3, x4, x5)  =  U20_GAA(x5)
U21_GAA(x1, x2, x3, x4, x5)  =  U21_GAA(x2, x5)
HIGHER_VALUED102_IN_AG(x1, x2)  =  HIGHER_VALUED102_IN_AG(x2)
U35_AG(x1, x2, x3)  =  U35_AG(x3)
U22_GAA(x1, x2, x3, x4, x5)  =  U22_GAA(x5)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x2, x5)
HIGHER_VALUED107_IN_AG(x1, x2)  =  HIGHER_VALUED107_IN_AG(x2)
U36_AG(x1, x2, x3)  =  U36_AG(x3)
U24_GAA(x1, x2, x3, x4, x5)  =  U24_GAA(x5)
U25_GAA(x1, x2, x3, x4, x5)  =  U25_GAA(x2, x5)
HIGHER_VALUED112_IN_AG(x1, x2)  =  HIGHER_VALUED112_IN_AG(x2)
U37_AG(x1, x2, x3)  =  U37_AG(x3)
U26_GAA(x1, x2, x3, x4, x5)  =  U26_GAA(x5)
U27_GAA(x1, x2, x3, x4, x5)  =  U27_GAA(x2, x5)
HIGHER_VALUED118_IN_AG(x1, x2)  =  HIGHER_VALUED118_IN_AG(x2)
U38_AG(x1, x2, x3)  =  U38_AG(x3)
U28_GAA(x1, x2, x3, x4, x5)  =  U28_GAA(x5)
U40_GA(x1, x2, x3, x4)  =  U40_GA(x4)
U41_GA(x1, x2, x3, x4)  =  U41_GA(x4)
U42_GA(x1, x2, x3, x4)  =  U42_GA(x4)
U43_GA(x1, x2, x3, x4)  =  U43_GA(x4)
U44_GA(x1, x2, x3, x4)  =  U44_GA(x4)
U45_GA(x1, x2, x3, x4)  =  U45_GA(x4)
U46_GA(x1, x2, x3, x4)  =  U46_GA(x4)
U47_GA(x1, x2, x3, x4, x5)  =  U47_GA(x3, x5)
U48_GA(x1, x2, x3, x4, x5)  =  U48_GA(x5)
U49_GA(x1, x2, x3, x4, x5)  =  U49_GA(x3, x5)
U50_GA(x1, x2, x3, x4, x5)  =  U50_GA(x5)
U51_GA(x1, x2, x3, x4, x5)  =  U51_GA(x1, x3, x5)
U52_GA(x1, x2, x3, x4, x5)  =  U52_GA(x5)
U53_GA(x1, x2, x3, x4, x5)  =  U53_GA(x1, x3, x5)
U54_GA(x1, x2, x3, x4, x5)  =  U54_GA(x5)
U55_GA(x1, x2, x3, x4, x5)  =  U55_GA(x1, x3, x5)
U56_GA(x1, x2, x3, x4, x5)  =  U56_GA(x5)
U57_GA(x1, x2, x3, x4, x5)  =  U57_GA(x1, x3, x5)
U58_GA(x1, x2, x3, x4, x5)  =  U58_GA(x5)
U59_GA(x1, x2, x3, x4, x5)  =  U59_GA(x1, x3, x5)
U60_GA(x1, x2, x3, x4, x5)  =  U60_GA(x5)
U61_GA(x1, x2, x3, x4, x5)  =  U61_GA(x1, x3, x5)
U62_GA(x1, x2, x3, x4, x5)  =  U62_GA(x5)
U63_GA(x1, x2, x3, x4, x5)  =  U63_GA(x1, x3, x5)
U64_GA(x1, x2, x3, x4, x5)  =  U64_GA(x5)
U65_GA(x1, x2, x3, x4, x5)  =  U65_GA(x1, x3, x5)
U66_GA(x1, x2, x3, x4, x5)  =  U66_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 149 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

GREATER64_IN_GA(s(T41), s(T42)) → GREATER64_IN_GA(T41, T42)

The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)
GREATER64_IN_GA(x1, x2)  =  GREATER64_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GREATER64_IN_GA(s(T41), s(T42)) → GREATER64_IN_GA(T41, T42)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GREATER64_IN_GA(x1, x2)  =  GREATER64_IN_GA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

GREATER64_IN_GA(s(T41)) → GREATER64_IN_GA(T41)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GREATER64_IN_GA(s(T41)) → GREATER64_IN_GA(T41)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

GREATER64_IN_AG(s(T41), s(T42)) → GREATER64_IN_AG(T41, T42)

The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)
GREATER64_IN_AG(x1, x2)  =  GREATER64_IN_AG(x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

GREATER64_IN_AG(s(T41), s(T42)) → GREATER64_IN_AG(T41, T42)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GREATER64_IN_AG(x1, x2)  =  GREATER64_IN_AG(x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

GREATER64_IN_AG(s(T42)) → GREATER64_IN_AG(T42)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GREATER64_IN_AG(s(T42)) → GREATER64_IN_AG(T42)
  The graph contains the following edges 1 > 1

(22) TRUE

(23) Obligation:

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

GREATER64_IN_GG(s(T41), s(T42)) → GREATER64_IN_GG(T41, T42)

The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)
GREATER64_IN_GG(x1, x2)  =  GREATER64_IN_GG(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

GREATER64_IN_GG(s(T41), s(T42)) → GREATER64_IN_GG(T41, T42)

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

(26) PiDPToQDPProof (EQUIVALENT transformation)

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

(27) Obligation:

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

GREATER64_IN_GG(s(T41), s(T42)) → GREATER64_IN_GG(T41, T42)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GREATER64_IN_GG(s(T41), s(T42)) → GREATER64_IN_GG(T41, T42)
  The graph contains the following edges 1 > 1, 2 > 2

(29) TRUE

(30) Obligation:

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

MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GGA(T37, T9, T38, T12, greater64_in_gg(T37, T38))
U9_GGA(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GGA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_GGA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GGA(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
U13_GGA(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GGA(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
U15_GGA(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GGA(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
U17_GGA(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GGA(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
U19_GGA(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82)), T84) → U21_GGA(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
U21_GGA(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → MAX_VALUED3_IN_GGA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89), T91) → U23_GGA(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
U23_GGA(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → MAX_VALUED3_IN_GGA(T88, s(T89), T91)
MAX_VALUED3_IN_GGA(.(T94, T95), T96, T98) → U25_GGA(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
U25_GGA(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → MAX_VALUED3_IN_GGA(T95, T96, T98)
MAX_VALUED3_IN_GGA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(T101, T102), T103, T105) → U27_GGA(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
U27_GGA(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → MAX_VALUED3_IN_GGA(T102, T103, T105)
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GAA(T37, T9, T38, T12, greater64_in_ga(T37, T38))
U9_GAA(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GAA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_GAA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GAA(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
U13_GAA(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → MAX_VALUED3_IN_GAA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GAA(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
U15_GAA(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → MAX_VALUED3_IN_GAA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GAA(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
U17_GAA(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → MAX_VALUED3_IN_GAA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GAA(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
U19_GAA(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → MAX_VALUED3_IN_GAA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GAA(.(s(T80), T81), s(s(T82)), T84) → U21_GAA(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
U21_GAA(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → MAX_VALUED3_IN_GAA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GAA(.(T87, T88), s(T89), T91) → U23_GAA(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
U23_GAA(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → MAX_VALUED3_IN_GAA(T88, s(T89), T91)
MAX_VALUED3_IN_GAA(.(T94, T95), T96, T98) → U25_GAA(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
U25_GAA(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → MAX_VALUED3_IN_GAA(T95, T96, T98)
MAX_VALUED3_IN_GAA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GAA(.(T101, T102), T103, T105) → U27_GAA(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
U27_GAA(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → MAX_VALUED3_IN_GAA(T102, T103, T105)

The TRS R consists of the following rules:

max_valued1_in_ga(.(T7, []), T7) → max_valued1_out_ga(.(T7, []), T7)
max_valued1_in_ga(.(0, .(T15, T9)), T12) → U39_ga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga([], T7, T7) → max_valued3_out_gga([], T7, T7)
max_valued3_in_gga(.(T15, T9), 0, T12) → U1_gga(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(T18), T9), s(0), T12) → U2_gga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(T21)), T9), s(s(0)), T12) → U3_gga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
max_valued3_in_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
max_valued3_in_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
max_valued3_in_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
max_valued3_in_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gga(T37, T9, T38, T12, greater64_in_gg(T37, T38))
greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gga(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → U10_gga(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa([], T7, T7) → max_valued3_out_gaa([], T7, T7)
max_valued3_in_gaa(.(T15, T9), 0, T12) → U1_gaa(T15, T9, T12, max_valued3_in_gga(T9, T15, T12))
max_valued3_in_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gga(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gga(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gga(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(T18), T9), s(0), T12) → U2_gaa(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
max_valued3_in_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gga(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gga(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → U14_gga(T52, T53, T54, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gga(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gga(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → U16_gga(T59, T60, T61, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gga(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gga(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → U18_gga(T66, T67, T68, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gga(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gga(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gga(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → U20_gga(T73, T74, T75, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
max_valued3_in_gga(.(s(T80), T81), s(s(T82)), T84) → U21_gga(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U21_gga(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → U22_gga(T80, T81, T82, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
max_valued3_in_gga(.(T87, T88), s(T89), T91) → U23_gga(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U23_gga(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → U24_gga(T87, T88, T89, T91, max_valued3_in_gga(T88, s(T89), T91))
max_valued3_in_gga(.(T94, T95), T96, T98) → U25_gga(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U25_gga(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → U26_gga(T94, T95, T96, T98, max_valued3_in_gga(T95, T96, T98))
max_valued3_in_gga(.(T101, T102), T103, T105) → U27_gga(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U27_gga(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → U28_gga(T101, T102, T103, T105, max_valued3_in_gga(T102, T103, T105))
U28_gga(T101, T102, T103, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued3_out_gga(.(T101, T102), T103, T105)
U26_gga(T94, T95, T96, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued3_out_gga(.(T94, T95), T96, T98)
U24_gga(T87, T88, T89, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued3_out_gga(.(T87, T88), s(T89), T91)
U22_gga(T80, T81, T82, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued3_out_gga(.(s(T80), T81), s(s(T82)), T84)
U20_gga(T73, T74, T75, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued3_out_gga(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gga(T66, T67, T68, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gga(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gga(T59, T60, T61, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gga(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gga(T52, T53, T54, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gga(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U2_gaa(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gaa(.(s(T18), T9), s(0), T12)
max_valued3_in_gaa(.(s(s(T21)), T9), s(s(0)), T12) → U3_gaa(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U3_gaa(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gaa(.(s(s(T21)), T9), s(s(0)), T12)
max_valued3_in_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12) → U4_gaa(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U4_gaa(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gaa(.(s(s(s(T24))), T9), s(s(s(0))), T12)
max_valued3_in_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → U5_gaa(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U5_gaa(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gaa(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → U6_gaa(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U6_gaa(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → U7_gaa(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U7_gaa(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → U8_gaa(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U8_gaa(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
max_valued3_in_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_gaa(T37, T9, T38, T12, greater64_in_ga(T37, T38))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U9_gaa(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → U10_gaa(T37, T9, T38, T12, max_valued3_in_gaa(T9, s(T39), T12))
max_valued3_in_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_gaa(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U11_gaa(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → U12_gaa(T43, T44, T45, T47, max_valued3_in_gaa(T44, s(T49), T47))
max_valued3_in_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_gaa(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U13_gaa(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → U14_gaa(T52, T53, T54, T56, max_valued3_in_gaa(T53, s(s(s(s(s(s(T54)))))), T56))
max_valued3_in_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_gaa(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U15_gaa(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → U16_gaa(T59, T60, T61, T63, max_valued3_in_gaa(T60, s(s(s(s(s(T61))))), T63))
max_valued3_in_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_gaa(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U17_gaa(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → U18_gaa(T66, T67, T68, T70, max_valued3_in_gaa(T67, s(s(s(s(T68)))), T70))
max_valued3_in_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_gaa(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U19_gaa(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → U20_gaa(T73, T74, T75, T77, max_valued3_in_gaa(T74, s(s(s(T75))), T77))
max_valued3_in_gaa(.(s(T80), T81), s(s(T82)), T84) → U21_gaa(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U21_gaa(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → U22_gaa(T80, T81, T82, T84, max_valued3_in_gaa(T81, s(s(T82)), T84))
max_valued3_in_gaa(.(T87, T88), s(T89), T91) → U23_gaa(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U23_gaa(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → U24_gaa(T87, T88, T89, T91, max_valued3_in_gaa(T88, s(T89), T91))
max_valued3_in_gaa(.(T94, T95), T96, T98) → U25_gaa(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U25_gaa(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → U26_gaa(T94, T95, T96, T98, max_valued3_in_gaa(T95, T96, T98))
max_valued3_in_gaa(.(T101, T102), T103, T105) → U27_gaa(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U27_gaa(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → U28_gaa(T101, T102, T103, T105, max_valued3_in_gaa(T102, T103, T105))
U28_gaa(T101, T102, T103, T105, max_valued3_out_gaa(T102, T103, T105)) → max_valued3_out_gaa(.(T101, T102), T103, T105)
U26_gaa(T94, T95, T96, T98, max_valued3_out_gaa(T95, T96, T98)) → max_valued3_out_gaa(.(T94, T95), T96, T98)
U24_gaa(T87, T88, T89, T91, max_valued3_out_gaa(T88, s(T89), T91)) → max_valued3_out_gaa(.(T87, T88), s(T89), T91)
U22_gaa(T80, T81, T82, T84, max_valued3_out_gaa(T81, s(s(T82)), T84)) → max_valued3_out_gaa(.(s(T80), T81), s(s(T82)), T84)
U20_gaa(T73, T74, T75, T77, max_valued3_out_gaa(T74, s(s(s(T75))), T77)) → max_valued3_out_gaa(.(s(s(T73)), T74), s(s(s(T75))), T77)
U18_gaa(T66, T67, T68, T70, max_valued3_out_gaa(T67, s(s(s(s(T68)))), T70)) → max_valued3_out_gaa(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70)
U16_gaa(T59, T60, T61, T63, max_valued3_out_gaa(T60, s(s(s(s(s(T61))))), T63)) → max_valued3_out_gaa(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63)
U14_gaa(T52, T53, T54, T56, max_valued3_out_gaa(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued3_out_gaa(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56)
U12_gaa(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gaa(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U10_gaa(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gaa(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U12_gga(T43, T44, T45, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued3_out_gga(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47)
U1_gaa(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gaa(.(T15, T9), 0, T12)
U10_gga(T37, T9, T38, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12)
U8_gga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12)
U7_gga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12)
U6_gga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued3_out_gga(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12)
U5_gga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued3_out_gga(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12)
U4_gga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued3_out_gga(.(s(s(s(T24))), T9), s(s(s(0))), T12)
U3_gga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued3_out_gga(.(s(s(T21)), T9), s(s(0)), T12)
U2_gga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued3_out_gga(.(s(T18), T9), s(0), T12)
U1_gga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued3_out_gga(.(T15, T9), 0, T12)
U39_ga(T15, T9, T12, max_valued3_out_gga(T9, T15, T12)) → max_valued1_out_ga(.(0, .(T15, T9)), T12)
max_valued1_in_ga(.(s(0), .(s(T18), T9)), T12) → U40_ga(T18, T9, T12, max_valued3_in_gga(T9, s(T18), T12))
U40_ga(T18, T9, T12, max_valued3_out_gga(T9, s(T18), T12)) → max_valued1_out_ga(.(s(0), .(s(T18), T9)), T12)
max_valued1_in_ga(.(s(s(0)), .(s(s(T21)), T9)), T12) → U41_ga(T21, T9, T12, max_valued3_in_gga(T9, s(s(T21)), T12))
U41_ga(T21, T9, T12, max_valued3_out_gga(T9, s(s(T21)), T12)) → max_valued1_out_ga(.(s(s(0)), .(s(s(T21)), T9)), T12)
max_valued1_in_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12) → U42_ga(T24, T9, T12, max_valued3_in_gga(T9, s(s(s(T24))), T12))
U42_ga(T24, T9, T12, max_valued3_out_gga(T9, s(s(s(T24))), T12)) → max_valued1_out_ga(.(s(s(s(0))), .(s(s(s(T24))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12) → U43_ga(T27, T9, T12, max_valued3_in_gga(T9, s(s(s(s(T27)))), T12))
U43_ga(T27, T9, T12, max_valued3_out_gga(T9, s(s(s(s(T27)))), T12)) → max_valued1_out_ga(.(s(s(s(s(0)))), .(s(s(s(s(T27)))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12) → U44_ga(T30, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(T30))))), T12))
U44_ga(T30, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(T30))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(0))))), .(s(s(s(s(s(T30))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12) → U45_ga(T33, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(T33)))))), T12))
U45_ga(T33, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(T33)))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(T33)))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12) → U46_ga(T36, T9, T12, max_valued3_in_gga(T9, s(s(s(s(s(s(s(T36))))))), T12))
U46_ga(T36, T9, T12, max_valued3_out_gga(T9, s(s(s(s(s(s(s(T36))))))), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(T36))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12) → U47_ga(T38, T37, T9, T12, greater64_in_gg(T37, T38))
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
U47_ga(T38, T37, T9, T12, greater64_out_gg(T37, T38)) → U48_ga(T38, T37, T9, T12, max_valued3_in_gaa(T9, s(T39), T12))
U48_ga(T38, T37, T9, T12, max_valued3_out_gaa(T9, s(T39), T12)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(s(T38)))))))), .(s(s(s(s(s(s(s(T37))))))), T9)), T12)
max_valued1_in_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47) → U49_ga(T45, T43, T44, T47, higher_valued77_in_ag(T48, T43))
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
U49_ga(T45, T43, T44, T47, higher_valued77_out_ag(T48, T43)) → U50_ga(T45, T43, T44, T47, max_valued3_in_gaa(T44, s(T49), T47))
U50_ga(T45, T43, T44, T47, max_valued3_out_gaa(T44, s(T49), T47)) → max_valued1_out_ga(.(s(s(s(s(s(s(s(T45))))))), .(s(s(s(s(s(s(T43)))))), T44)), T47)
max_valued1_in_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56) → U51_ga(T54, T52, T53, T56, higher_valued82_in_gg(T54, T52))
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
U51_ga(T54, T52, T53, T56, higher_valued82_out_gg(T54, T52)) → U52_ga(T54, T52, T53, T56, max_valued3_in_gga(T53, s(s(s(s(s(s(T54)))))), T56))
U52_ga(T54, T52, T53, T56, max_valued3_out_gga(T53, s(s(s(s(s(s(T54)))))), T56)) → max_valued1_out_ga(.(s(s(s(s(s(s(T54)))))), .(s(s(s(s(s(T52))))), T53)), T56)
max_valued1_in_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63) → U53_ga(T61, T59, T60, T63, higher_valued87_in_gg(T61, T59))
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
U53_ga(T61, T59, T60, T63, higher_valued87_out_gg(T61, T59)) → U54_ga(T61, T59, T60, T63, max_valued3_in_gga(T60, s(s(s(s(s(T61))))), T63))
U54_ga(T61, T59, T60, T63, max_valued3_out_gga(T60, s(s(s(s(s(T61))))), T63)) → max_valued1_out_ga(.(s(s(s(s(s(T61))))), .(s(s(s(s(T59)))), T60)), T63)
max_valued1_in_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70) → U55_ga(T68, T66, T67, T70, higher_valued92_in_gg(T68, T66))
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
U55_ga(T68, T66, T67, T70, higher_valued92_out_gg(T68, T66)) → U56_ga(T68, T66, T67, T70, max_valued3_in_gga(T67, s(s(s(s(T68)))), T70))
U56_ga(T68, T66, T67, T70, max_valued3_out_gga(T67, s(s(s(s(T68)))), T70)) → max_valued1_out_ga(.(s(s(s(s(T68)))), .(s(s(s(T66))), T67)), T70)
max_valued1_in_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77) → U57_ga(T75, T73, T74, T77, higher_valued97_in_gg(T75, T73))
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
U57_ga(T75, T73, T74, T77, higher_valued97_out_gg(T75, T73)) → U58_ga(T75, T73, T74, T77, max_valued3_in_gga(T74, s(s(s(T75))), T77))
U58_ga(T75, T73, T74, T77, max_valued3_out_gga(T74, s(s(s(T75))), T77)) → max_valued1_out_ga(.(s(s(s(T75))), .(s(s(T73)), T74)), T77)
max_valued1_in_ga(.(s(s(T82)), .(s(T80), T81)), T84) → U59_ga(T82, T80, T81, T84, higher_valued102_in_gg(T82, T80))
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
U59_ga(T82, T80, T81, T84, higher_valued102_out_gg(T82, T80)) → U60_ga(T82, T80, T81, T84, max_valued3_in_gga(T81, s(s(T82)), T84))
U60_ga(T82, T80, T81, T84, max_valued3_out_gga(T81, s(s(T82)), T84)) → max_valued1_out_ga(.(s(s(T82)), .(s(T80), T81)), T84)
max_valued1_in_ga(.(s(T89), .(T87, T88)), T91) → U61_ga(T89, T87, T88, T91, higher_valued107_in_gg(T89, T87))
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
U61_ga(T89, T87, T88, T91, higher_valued107_out_gg(T89, T87)) → U62_ga(T89, T87, T88, T91, max_valued3_in_gga(T88, s(T89), T91))
U62_ga(T89, T87, T88, T91, max_valued3_out_gga(T88, s(T89), T91)) → max_valued1_out_ga(.(s(T89), .(T87, T88)), T91)
max_valued1_in_ga(.(T96, .(T94, T95)), T98) → U63_ga(T96, T94, T95, T98, higher_valued112_in_gg(T96, T94))
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
U63_ga(T96, T94, T95, T98, higher_valued112_out_gg(T96, T94)) → U64_ga(T96, T94, T95, T98, max_valued3_in_gga(T95, T96, T98))
U64_ga(T96, T94, T95, T98, max_valued3_out_gga(T95, T96, T98)) → max_valued1_out_ga(.(T96, .(T94, T95)), T98)
max_valued1_in_ga(.(T103, .(T101, T102)), T105) → U65_ga(T103, T101, T102, T105, higher_valued118_in_gg(T103, T101))
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)
U65_ga(T103, T101, T102, T105, higher_valued118_out_gg(T103, T101)) → U66_ga(T103, T101, T102, T105, max_valued3_in_gga(T102, T103, T105))
U66_ga(T103, T101, T102, T105, max_valued3_out_gga(T102, T103, T105)) → max_valued1_out_ga(.(T103, .(T101, T102)), T105)

The argument filtering Pi contains the following mapping:
max_valued1_in_ga(x1, x2)  =  max_valued1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
max_valued1_out_ga(x1, x2)  =  max_valued1_out_ga
0  =  0
U39_ga(x1, x2, x3, x4)  =  U39_ga(x4)
max_valued3_in_gga(x1, x2, x3)  =  max_valued3_in_gga(x1, x2)
max_valued3_out_gga(x1, x2, x3)  =  max_valued3_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x2, x5)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x5)
max_valued3_in_gaa(x1, x2, x3)  =  max_valued3_in_gaa(x1)
max_valued3_out_gaa(x1, x2, x3)  =  max_valued3_out_gaa
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x2, x5)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x2, x3, x5)
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x5)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x2, x3, x5)
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x5)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x2, x3, x5)
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x2, x3, x5)
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x2, x3, x5)
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x5)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x2, x3, x5)
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U25_gga(x1, x2, x3, x4, x5)  =  U25_gga(x2, x3, x5)
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x5)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x2, x3, x5)
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x5)
U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x4)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x4)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x4)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
U10_gaa(x1, x2, x3, x4, x5)  =  U10_gaa(x5)
U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x2, x5)
U12_gaa(x1, x2, x3, x4, x5)  =  U12_gaa(x5)
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x2, x5)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x5)
U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
U16_gaa(x1, x2, x3, x4, x5)  =  U16_gaa(x5)
U17_gaa(x1, x2, x3, x4, x5)  =  U17_gaa(x2, x5)
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x5)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x2, x5)
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
U20_gaa(x1, x2, x3, x4, x5)  =  U20_gaa(x5)
U21_gaa(x1, x2, x3, x4, x5)  =  U21_gaa(x2, x5)
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
U22_gaa(x1, x2, x3, x4, x5)  =  U22_gaa(x5)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x2, x5)
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
U24_gaa(x1, x2, x3, x4, x5)  =  U24_gaa(x5)
U25_gaa(x1, x2, x3, x4, x5)  =  U25_gaa(x2, x5)
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x2, x5)
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U40_ga(x1, x2, x3, x4)  =  U40_ga(x4)
U41_ga(x1, x2, x3, x4)  =  U41_ga(x4)
U42_ga(x1, x2, x3, x4)  =  U42_ga(x4)
U43_ga(x1, x2, x3, x4)  =  U43_ga(x4)
U44_ga(x1, x2, x3, x4)  =  U44_ga(x4)
U45_ga(x1, x2, x3, x4)  =  U45_ga(x4)
U46_ga(x1, x2, x3, x4)  =  U46_ga(x4)
U47_ga(x1, x2, x3, x4, x5)  =  U47_ga(x3, x5)
U48_ga(x1, x2, x3, x4, x5)  =  U48_ga(x5)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x5)
U51_ga(x1, x2, x3, x4, x5)  =  U51_ga(x1, x3, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x5)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x5)
U55_ga(x1, x2, x3, x4, x5)  =  U55_ga(x1, x3, x5)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x5)
U57_ga(x1, x2, x3, x4, x5)  =  U57_ga(x1, x3, x5)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x3, x5)
U60_ga(x1, x2, x3, x4, x5)  =  U60_ga(x5)
U61_ga(x1, x2, x3, x4, x5)  =  U61_ga(x1, x3, x5)
U62_ga(x1, x2, x3, x4, x5)  =  U62_ga(x5)
U63_ga(x1, x2, x3, x4, x5)  =  U63_ga(x1, x3, x5)
U64_ga(x1, x2, x3, x4, x5)  =  U64_ga(x5)
U65_ga(x1, x2, x3, x4, x5)  =  U65_ga(x1, x3, x5)
U66_ga(x1, x2, x3, x4, x5)  =  U66_ga(x5)
MAX_VALUED3_IN_GGA(x1, x2, x3)  =  MAX_VALUED3_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x2, x5)
MAX_VALUED3_IN_GAA(x1, x2, x3)  =  MAX_VALUED3_IN_GAA(x1)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x2, x5)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x2, x3, x5)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x2, x3, x5)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x2, x3, x5)
U21_GGA(x1, x2, x3, x4, x5)  =  U21_GGA(x2, x3, x5)
U23_GGA(x1, x2, x3, x4, x5)  =  U23_GGA(x2, x3, x5)
U25_GGA(x1, x2, x3, x4, x5)  =  U25_GGA(x2, x3, x5)
U27_GGA(x1, x2, x3, x4, x5)  =  U27_GGA(x2, x3, x5)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x2, x5)
U11_GAA(x1, x2, x3, x4, x5)  =  U11_GAA(x2, x5)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x2, x5)
U15_GAA(x1, x2, x3, x4, x5)  =  U15_GAA(x2, x5)
U17_GAA(x1, x2, x3, x4, x5)  =  U17_GAA(x2, x5)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x2, x5)
U21_GAA(x1, x2, x3, x4, x5)  =  U21_GAA(x2, x5)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x2, x5)
U25_GAA(x1, x2, x3, x4, x5)  =  U25_GAA(x2, x5)
U27_GAA(x1, x2, x3, x4, x5)  =  U27_GAA(x2, x5)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GGA(T37, T9, T38, T12, greater64_in_gg(T37, T38))
U9_GGA(T37, T9, T38, T12, greater64_out_gg(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(s(T18), T9), s(0), T12) → MAX_VALUED3_IN_GGA(T9, s(T18), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GGA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_GGA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9), s(s(0)), T12) → MAX_VALUED3_IN_GGA(T9, s(s(T21)), T12)
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GGA(T52, T53, T54, T56, higher_valued82_in_gg(T54, T52))
U13_GGA(T52, T53, T54, T56, higher_valued82_out_gg(T54, T52)) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GGA(T59, T60, T61, T63, higher_valued87_in_gg(T61, T59))
U15_GGA(T59, T60, T61, T63, higher_valued87_out_gg(T61, T59)) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GGA(T66, T67, T68, T70, higher_valued92_in_gg(T68, T66))
U17_GGA(T66, T67, T68, T70, higher_valued92_out_gg(T68, T66)) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GGA(T73, T74, T75, T77, higher_valued97_in_gg(T75, T73))
U19_GGA(T73, T74, T75, T77, higher_valued97_out_gg(T75, T73)) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82)), T84) → U21_GGA(T80, T81, T82, T84, higher_valued102_in_gg(T82, T80))
U21_GGA(T80, T81, T82, T84, higher_valued102_out_gg(T82, T80)) → MAX_VALUED3_IN_GGA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89), T91) → U23_GGA(T87, T88, T89, T91, higher_valued107_in_gg(T89, T87))
U23_GGA(T87, T88, T89, T91, higher_valued107_out_gg(T89, T87)) → MAX_VALUED3_IN_GGA(T88, s(T89), T91)
MAX_VALUED3_IN_GGA(.(T94, T95), T96, T98) → U25_GGA(T94, T95, T96, T98, higher_valued112_in_gg(T96, T94))
U25_GGA(T94, T95, T96, T98, higher_valued112_out_gg(T96, T94)) → MAX_VALUED3_IN_GGA(T95, T96, T98)
MAX_VALUED3_IN_GGA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GGA(.(T101, T102), T103, T105) → U27_GGA(T101, T102, T103, T105, higher_valued118_in_gg(T103, T101))
U27_GGA(T101, T102, T103, T105, higher_valued118_out_gg(T103, T101)) → MAX_VALUED3_IN_GGA(T102, T103, T105)
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9), s(s(s(0))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9), s(s(s(s(0)))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0)))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0))))))), T12) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38)))))))), T12) → U9_GAA(T37, T9, T38, T12, greater64_in_ga(T37, T38))
U9_GAA(T37, T9, T38, T12, greater64_out_ga(T37, T38)) → MAX_VALUED3_IN_GAA(T9, s(T39), T12)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45))))))), T47) → U11_GAA(T43, T44, T45, T47, higher_valued77_in_ag(T48, T43))
U11_GAA(T43, T44, T45, T47, higher_valued77_out_ag(T48, T43)) → MAX_VALUED3_IN_GAA(T44, s(T49), T47)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54)))))), T56) → U13_GAA(T52, T53, T54, T56, higher_valued82_in_ag(T54, T52))
U13_GAA(T52, T53, T54, T56, higher_valued82_out_ag(T54, T52)) → MAX_VALUED3_IN_GAA(T53, s(s(s(s(s(s(T54)))))), T56)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61))))), T63) → U15_GAA(T59, T60, T61, T63, higher_valued87_in_ag(T61, T59))
U15_GAA(T59, T60, T61, T63, higher_valued87_out_ag(T61, T59)) → MAX_VALUED3_IN_GAA(T60, s(s(s(s(s(T61))))), T63)
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67), s(s(s(s(T68)))), T70) → U17_GAA(T66, T67, T68, T70, higher_valued92_in_ag(T68, T66))
U17_GAA(T66, T67, T68, T70, higher_valued92_out_ag(T68, T66)) → MAX_VALUED3_IN_GAA(T67, s(s(s(s(T68)))), T70)
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74), s(s(s(T75))), T77) → U19_GAA(T73, T74, T75, T77, higher_valued97_in_ag(T75, T73))
U19_GAA(T73, T74, T75, T77, higher_valued97_out_ag(T75, T73)) → MAX_VALUED3_IN_GAA(T74, s(s(s(T75))), T77)
MAX_VALUED3_IN_GAA(.(s(T80), T81), s(s(T82)), T84) → U21_GAA(T80, T81, T82, T84, higher_valued102_in_ag(T82, T80))
U21_GAA(T80, T81, T82, T84, higher_valued102_out_ag(T82, T80)) → MAX_VALUED3_IN_GAA(T81, s(s(T82)), T84)
MAX_VALUED3_IN_GAA(.(T87, T88), s(T89), T91) → U23_GAA(T87, T88, T89, T91, higher_valued107_in_ag(T89, T87))
U23_GAA(T87, T88, T89, T91, higher_valued107_out_ag(T89, T87)) → MAX_VALUED3_IN_GAA(T88, s(T89), T91)
MAX_VALUED3_IN_GAA(.(T94, T95), T96, T98) → U25_GAA(T94, T95, T96, T98, higher_valued112_in_ag(T96, T94))
U25_GAA(T94, T95, T96, T98, higher_valued112_out_ag(T96, T94)) → MAX_VALUED3_IN_GAA(T95, T96, T98)
MAX_VALUED3_IN_GAA(.(T15, T9), 0, T12) → MAX_VALUED3_IN_GGA(T9, T15, T12)
MAX_VALUED3_IN_GAA(.(T101, T102), T103, T105) → U27_GAA(T101, T102, T103, T105, higher_valued118_in_ag(T103, T101))
U27_GAA(T101, T102, T103, T105, higher_valued118_out_ag(T103, T101)) → MAX_VALUED3_IN_GAA(T102, T103, T105)

The TRS R consists of the following rules:

greater64_in_gg(s(T40), 0) → greater64_out_gg(s(T40), 0)
greater64_in_gg(s(T41), s(T42)) → U29_gg(T41, T42, greater64_in_gg(T41, T42))
higher_valued77_in_ag(T50, T51) → U30_ag(T50, T51, greater64_in_ag(s(s(T50)), s(s(s(s(s(s(T51))))))))
higher_valued82_in_gg(T57, T58) → U31_gg(T57, T58, greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
higher_valued87_in_gg(T64, T65) → U32_gg(T64, T65, greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
higher_valued92_in_gg(T71, T72) → U33_gg(T71, T72, greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
higher_valued97_in_gg(T78, T79) → U34_gg(T78, T79, greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
higher_valued102_in_gg(T85, T86) → U35_gg(T85, T86, greater64_in_gg(s(s(s(T85))), s(T86)))
higher_valued107_in_gg(T92, T93) → U36_gg(T92, T93, greater64_in_gg(s(s(T92)), T93))
higher_valued112_in_gg(T99, T100) → U37_gg(T99, T100, greater64_in_gg(s(T99), T100))
higher_valued118_in_gg(T106, T107) → U38_gg(T106, T107, greater64_in_gg(s(T106), T107))
greater64_in_ga(s(T40), 0) → greater64_out_ga(s(T40), 0)
greater64_in_ga(s(T41), s(T42)) → U29_ga(T41, T42, greater64_in_ga(T41, T42))
higher_valued82_in_ag(T57, T58) → U31_ag(T57, T58, greater64_in_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
higher_valued87_in_ag(T64, T65) → U32_ag(T64, T65, greater64_in_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
higher_valued92_in_ag(T71, T72) → U33_ag(T71, T72, greater64_in_ag(s(s(s(s(s(T71))))), s(s(s(T72)))))
higher_valued97_in_ag(T78, T79) → U34_ag(T78, T79, greater64_in_ag(s(s(s(s(T78)))), s(s(T79))))
higher_valued102_in_ag(T85, T86) → U35_ag(T85, T86, greater64_in_ag(s(s(s(T85))), s(T86)))
higher_valued107_in_ag(T92, T93) → U36_ag(T92, T93, greater64_in_ag(s(s(T92)), T93))
higher_valued112_in_ag(T99, T100) → U37_ag(T99, T100, greater64_in_ag(s(T99), T100))
higher_valued118_in_ag(T106, T107) → U38_ag(T106, T107, greater64_in_ag(s(T106), T107))
U29_gg(T41, T42, greater64_out_gg(T41, T42)) → greater64_out_gg(s(T41), s(T42))
U30_ag(T50, T51, greater64_out_ag(s(s(T50)), s(s(s(s(s(s(T51)))))))) → higher_valued77_out_ag(T50, T51)
U31_gg(T57, T58, greater64_out_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_gg(T57, T58)
U32_gg(T64, T65, greater64_out_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_gg(T64, T65)
U33_gg(T71, T72, greater64_out_gg(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_gg(T71, T72)
U34_gg(T78, T79, greater64_out_gg(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_gg(T78, T79)
U35_gg(T85, T86, greater64_out_gg(s(s(s(T85))), s(T86))) → higher_valued102_out_gg(T85, T86)
U36_gg(T92, T93, greater64_out_gg(s(s(T92)), T93)) → higher_valued107_out_gg(T92, T93)
U37_gg(T99, T100, greater64_out_gg(s(T99), T100)) → higher_valued112_out_gg(T99, T100)
U38_gg(T106, T107, greater64_out_gg(s(T106), T107)) → higher_valued118_out_gg(T106, T107)
U29_ga(T41, T42, greater64_out_ga(T41, T42)) → greater64_out_ga(s(T41), s(T42))
U31_ag(T57, T58, greater64_out_ag(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58))))))) → higher_valued82_out_ag(T57, T58)
U32_ag(T64, T65, greater64_out_ag(s(s(s(s(s(s(T64)))))), s(s(s(s(T65)))))) → higher_valued87_out_ag(T64, T65)
U33_ag(T71, T72, greater64_out_ag(s(s(s(s(s(T71))))), s(s(s(T72))))) → higher_valued92_out_ag(T71, T72)
U34_ag(T78, T79, greater64_out_ag(s(s(s(s(T78)))), s(s(T79)))) → higher_valued97_out_ag(T78, T79)
U35_ag(T85, T86, greater64_out_ag(s(s(s(T85))), s(T86))) → higher_valued102_out_ag(T85, T86)
U36_ag(T92, T93, greater64_out_ag(s(s(T92)), T93)) → higher_valued107_out_ag(T92, T93)
U37_ag(T99, T100, greater64_out_ag(s(T99), T100)) → higher_valued112_out_ag(T99, T100)
U38_ag(T106, T107, greater64_out_ag(s(T106), T107)) → higher_valued118_out_ag(T106, T107)
greater64_in_ag(s(T41), s(T42)) → U29_ag(T41, T42, greater64_in_ag(T41, T42))
greater64_in_ag(s(T40), 0) → greater64_out_ag(s(T40), 0)
U29_ag(T41, T42, greater64_out_ag(T41, T42)) → greater64_out_ag(s(T41), s(T42))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
greater64_in_gg(x1, x2)  =  greater64_in_gg(x1, x2)
greater64_out_gg(x1, x2)  =  greater64_out_gg
U29_gg(x1, x2, x3)  =  U29_gg(x3)
higher_valued77_in_ag(x1, x2)  =  higher_valued77_in_ag(x2)
U30_ag(x1, x2, x3)  =  U30_ag(x3)
greater64_in_ag(x1, x2)  =  greater64_in_ag(x2)
greater64_out_ag(x1, x2)  =  greater64_out_ag
U29_ag(x1, x2, x3)  =  U29_ag(x3)
higher_valued77_out_ag(x1, x2)  =  higher_valued77_out_ag
higher_valued82_in_gg(x1, x2)  =  higher_valued82_in_gg(x1, x2)
U31_gg(x1, x2, x3)  =  U31_gg(x3)
higher_valued82_out_gg(x1, x2)  =  higher_valued82_out_gg
higher_valued87_in_gg(x1, x2)  =  higher_valued87_in_gg(x1, x2)
U32_gg(x1, x2, x3)  =  U32_gg(x3)
higher_valued87_out_gg(x1, x2)  =  higher_valued87_out_gg
higher_valued92_in_gg(x1, x2)  =  higher_valued92_in_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x3)
higher_valued92_out_gg(x1, x2)  =  higher_valued92_out_gg
higher_valued97_in_gg(x1, x2)  =  higher_valued97_in_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x3)
higher_valued97_out_gg(x1, x2)  =  higher_valued97_out_gg
higher_valued102_in_gg(x1, x2)  =  higher_valued102_in_gg(x1, x2)
U35_gg(x1, x2, x3)  =  U35_gg(x3)
higher_valued102_out_gg(x1, x2)  =  higher_valued102_out_gg
higher_valued107_in_gg(x1, x2)  =  higher_valued107_in_gg(x1, x2)
U36_gg(x1, x2, x3)  =  U36_gg(x3)
higher_valued107_out_gg(x1, x2)  =  higher_valued107_out_gg
higher_valued112_in_gg(x1, x2)  =  higher_valued112_in_gg(x1, x2)
U37_gg(x1, x2, x3)  =  U37_gg(x3)
higher_valued112_out_gg(x1, x2)  =  higher_valued112_out_gg
higher_valued118_in_gg(x1, x2)  =  higher_valued118_in_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x3)
higher_valued118_out_gg(x1, x2)  =  higher_valued118_out_gg
greater64_in_ga(x1, x2)  =  greater64_in_ga(x1)
greater64_out_ga(x1, x2)  =  greater64_out_ga(x2)
U29_ga(x1, x2, x3)  =  U29_ga(x3)
higher_valued82_in_ag(x1, x2)  =  higher_valued82_in_ag(x2)
U31_ag(x1, x2, x3)  =  U31_ag(x3)
higher_valued82_out_ag(x1, x2)  =  higher_valued82_out_ag
higher_valued87_in_ag(x1, x2)  =  higher_valued87_in_ag(x2)
U32_ag(x1, x2, x3)  =  U32_ag(x3)
higher_valued87_out_ag(x1, x2)  =  higher_valued87_out_ag
higher_valued92_in_ag(x1, x2)  =  higher_valued92_in_ag(x2)
U33_ag(x1, x2, x3)  =  U33_ag(x3)
higher_valued92_out_ag(x1, x2)  =  higher_valued92_out_ag
higher_valued97_in_ag(x1, x2)  =  higher_valued97_in_ag(x2)
U34_ag(x1, x2, x3)  =  U34_ag(x3)
higher_valued97_out_ag(x1, x2)  =  higher_valued97_out_ag
higher_valued102_in_ag(x1, x2)  =  higher_valued102_in_ag(x2)
U35_ag(x1, x2, x3)  =  U35_ag(x3)
higher_valued102_out_ag(x1, x2)  =  higher_valued102_out_ag
higher_valued107_in_ag(x1, x2)  =  higher_valued107_in_ag(x2)
U36_ag(x1, x2, x3)  =  U36_ag(x3)
higher_valued107_out_ag(x1, x2)  =  higher_valued107_out_ag
higher_valued112_in_ag(x1, x2)  =  higher_valued112_in_ag(x2)
U37_ag(x1, x2, x3)  =  U37_ag(x3)
higher_valued112_out_ag(x1, x2)  =  higher_valued112_out_ag
higher_valued118_in_ag(x1, x2)  =  higher_valued118_in_ag(x2)
U38_ag(x1, x2, x3)  =  U38_ag(x3)
higher_valued118_out_ag(x1, x2)  =  higher_valued118_out_ag
MAX_VALUED3_IN_GGA(x1, x2, x3)  =  MAX_VALUED3_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x2, x5)
MAX_VALUED3_IN_GAA(x1, x2, x3)  =  MAX_VALUED3_IN_GAA(x1)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x2, x5)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x2, x3, x5)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x2, x3, x5)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x2, x3, x5)
U21_GGA(x1, x2, x3, x4, x5)  =  U21_GGA(x2, x3, x5)
U23_GGA(x1, x2, x3, x4, x5)  =  U23_GGA(x2, x3, x5)
U25_GGA(x1, x2, x3, x4, x5)  =  U25_GGA(x2, x3, x5)
U27_GGA(x1, x2, x3, x4, x5)  =  U27_GGA(x2, x3, x5)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x2, x5)
U11_GAA(x1, x2, x3, x4, x5)  =  U11_GAA(x2, x5)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x2, x5)
U15_GAA(x1, x2, x3, x4, x5)  =  U15_GAA(x2, x5)
U17_GAA(x1, x2, x3, x4, x5)  =  U17_GAA(x2, x5)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x2, x5)
U21_GAA(x1, x2, x3, x4, x5)  =  U21_GAA(x2, x5)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x2, x5)
U25_GAA(x1, x2, x3, x4, x5)  =  U25_GAA(x2, x5)
U27_GAA(x1, x2, x3, x4, x5)  =  U27_GAA(x2, x5)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0)) → MAX_VALUED3_IN_GGA(T9, s(T18))
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0))) → MAX_VALUED3_IN_GGA(T9, s(s(T21)))
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0)))) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0)))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0))))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0)))))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38))))))))) → U9_GGA(T9, greater64_in_gg(T37, T38))
U9_GGA(T9, greater64_out_gg) → MAX_VALUED3_IN_GAA(T9)
MAX_VALUED3_IN_GAA(.(s(T18), T9)) → MAX_VALUED3_IN_GGA(T9, s(T18))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45)))))))) → U11_GGA(T44, higher_valued77_in_ag(T43))
U11_GGA(T44, higher_valued77_out_ag) → MAX_VALUED3_IN_GAA(T44)
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(T21)))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54))))))) → U13_GGA(T53, T54, higher_valued82_in_gg(T54, T52))
U13_GGA(T53, T54, higher_valued82_out_gg) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61)))))) → U15_GGA(T60, T61, higher_valued87_in_gg(T61, T59))
U15_GGA(T60, T61, higher_valued87_out_gg) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))))
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68))))) → U17_GGA(T67, T68, higher_valued92_in_gg(T68, T66))
U17_GGA(T67, T68, higher_valued92_out_gg) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))))
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75)))) → U19_GGA(T74, T75, higher_valued97_in_gg(T75, T73))
U19_GGA(T74, T75, higher_valued97_out_gg) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))))
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82))) → U21_GGA(T81, T82, higher_valued102_in_gg(T82, T80))
U21_GGA(T81, T82, higher_valued102_out_gg) → MAX_VALUED3_IN_GGA(T81, s(s(T82)))
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89)) → U23_GGA(T88, T89, higher_valued107_in_gg(T89, T87))
U23_GGA(T88, T89, higher_valued107_out_gg) → MAX_VALUED3_IN_GGA(T88, s(T89))
MAX_VALUED3_IN_GGA(.(T94, T95), T96) → U25_GGA(T95, T96, higher_valued112_in_gg(T96, T94))
U25_GGA(T95, T96, higher_valued112_out_gg) → MAX_VALUED3_IN_GGA(T95, T96)
MAX_VALUED3_IN_GGA(.(T15, T9), 0) → MAX_VALUED3_IN_GGA(T9, T15)
MAX_VALUED3_IN_GGA(.(T101, T102), T103) → U27_GGA(T102, T103, higher_valued118_in_gg(T103, T101))
U27_GGA(T102, T103, higher_valued118_out_gg) → MAX_VALUED3_IN_GGA(T102, T103)
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9)) → U9_GAA(T9, greater64_in_ga(T37))
U9_GAA(T9, greater64_out_ga(T38)) → MAX_VALUED3_IN_GAA(T9)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44)) → U11_GAA(T44, higher_valued77_in_ag(T43))
U11_GAA(T44, higher_valued77_out_ag) → MAX_VALUED3_IN_GAA(T44)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53)) → U13_GAA(T53, higher_valued82_in_ag(T52))
U13_GAA(T53, higher_valued82_out_ag) → MAX_VALUED3_IN_GAA(T53)
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60)) → U15_GAA(T60, higher_valued87_in_ag(T59))
U15_GAA(T60, higher_valued87_out_ag) → MAX_VALUED3_IN_GAA(T60)
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67)) → U17_GAA(T67, higher_valued92_in_ag(T66))
U17_GAA(T67, higher_valued92_out_ag) → MAX_VALUED3_IN_GAA(T67)
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74)) → U19_GAA(T74, higher_valued97_in_ag(T73))
U19_GAA(T74, higher_valued97_out_ag) → MAX_VALUED3_IN_GAA(T74)
MAX_VALUED3_IN_GAA(.(s(T80), T81)) → U21_GAA(T81, higher_valued102_in_ag(T80))
U21_GAA(T81, higher_valued102_out_ag) → MAX_VALUED3_IN_GAA(T81)
MAX_VALUED3_IN_GAA(.(T87, T88)) → U23_GAA(T88, higher_valued107_in_ag(T87))
U23_GAA(T88, higher_valued107_out_ag) → MAX_VALUED3_IN_GAA(T88)
MAX_VALUED3_IN_GAA(.(T94, T95)) → U25_GAA(T95, higher_valued112_in_ag(T94))
U25_GAA(T95, higher_valued112_out_ag) → MAX_VALUED3_IN_GAA(T95)
MAX_VALUED3_IN_GAA(.(T15, T9)) → MAX_VALUED3_IN_GGA(T9, T15)
MAX_VALUED3_IN_GAA(.(T101, T102)) → U27_GAA(T102, higher_valued118_in_ag(T101))
U27_GAA(T102, higher_valued118_out_ag) → MAX_VALUED3_IN_GAA(T102)

The TRS R consists of the following rules:

greater64_in_gg(s(T40), 0) → greater64_out_gg
greater64_in_gg(s(T41), s(T42)) → U29_gg(greater64_in_gg(T41, T42))
higher_valued77_in_ag(T51) → U30_ag(greater64_in_ag(s(s(s(s(s(s(T51))))))))
higher_valued82_in_gg(T57, T58) → U31_gg(greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
higher_valued87_in_gg(T64, T65) → U32_gg(greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
higher_valued92_in_gg(T71, T72) → U33_gg(greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
higher_valued97_in_gg(T78, T79) → U34_gg(greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
higher_valued102_in_gg(T85, T86) → U35_gg(greater64_in_gg(s(s(s(T85))), s(T86)))
higher_valued107_in_gg(T92, T93) → U36_gg(greater64_in_gg(s(s(T92)), T93))
higher_valued112_in_gg(T99, T100) → U37_gg(greater64_in_gg(s(T99), T100))
higher_valued118_in_gg(T106, T107) → U38_gg(greater64_in_gg(s(T106), T107))
greater64_in_ga(s(T40)) → greater64_out_ga(0)
greater64_in_ga(s(T41)) → U29_ga(greater64_in_ga(T41))
higher_valued82_in_ag(T58) → U31_ag(greater64_in_ag(s(s(s(s(s(T58)))))))
higher_valued87_in_ag(T65) → U32_ag(greater64_in_ag(s(s(s(s(T65))))))
higher_valued92_in_ag(T72) → U33_ag(greater64_in_ag(s(s(s(T72)))))
higher_valued97_in_ag(T79) → U34_ag(greater64_in_ag(s(s(T79))))
higher_valued102_in_ag(T86) → U35_ag(greater64_in_ag(s(T86)))
higher_valued107_in_ag(T93) → U36_ag(greater64_in_ag(T93))
higher_valued112_in_ag(T100) → U37_ag(greater64_in_ag(T100))
higher_valued118_in_ag(T107) → U38_ag(greater64_in_ag(T107))
U29_gg(greater64_out_gg) → greater64_out_gg
U30_ag(greater64_out_ag) → higher_valued77_out_ag
U31_gg(greater64_out_gg) → higher_valued82_out_gg
U32_gg(greater64_out_gg) → higher_valued87_out_gg
U33_gg(greater64_out_gg) → higher_valued92_out_gg
U34_gg(greater64_out_gg) → higher_valued97_out_gg
U35_gg(greater64_out_gg) → higher_valued102_out_gg
U36_gg(greater64_out_gg) → higher_valued107_out_gg
U37_gg(greater64_out_gg) → higher_valued112_out_gg
U38_gg(greater64_out_gg) → higher_valued118_out_gg
U29_ga(greater64_out_ga(T42)) → greater64_out_ga(s(T42))
U31_ag(greater64_out_ag) → higher_valued82_out_ag
U32_ag(greater64_out_ag) → higher_valued87_out_ag
U33_ag(greater64_out_ag) → higher_valued92_out_ag
U34_ag(greater64_out_ag) → higher_valued97_out_ag
U35_ag(greater64_out_ag) → higher_valued102_out_ag
U36_ag(greater64_out_ag) → higher_valued107_out_ag
U37_ag(greater64_out_ag) → higher_valued112_out_ag
U38_ag(greater64_out_ag) → higher_valued118_out_ag
greater64_in_ag(s(T42)) → U29_ag(greater64_in_ag(T42))
greater64_in_ag(0) → greater64_out_ag
U29_ag(greater64_out_ag) → greater64_out_ag

The set Q consists of the following terms:

greater64_in_gg(x0, x1)
higher_valued77_in_ag(x0)
higher_valued82_in_gg(x0, x1)
higher_valued87_in_gg(x0, x1)
higher_valued92_in_gg(x0, x1)
higher_valued97_in_gg(x0, x1)
higher_valued102_in_gg(x0, x1)
higher_valued107_in_gg(x0, x1)
higher_valued112_in_gg(x0, x1)
higher_valued118_in_gg(x0, x1)
greater64_in_ga(x0)
higher_valued82_in_ag(x0)
higher_valued87_in_ag(x0)
higher_valued92_in_ag(x0)
higher_valued97_in_ag(x0)
higher_valued102_in_ag(x0)
higher_valued107_in_ag(x0)
higher_valued112_in_ag(x0)
higher_valued118_in_ag(x0)
U29_gg(x0)
U30_ag(x0)
U31_gg(x0)
U32_gg(x0)
U33_gg(x0)
U34_gg(x0)
U35_gg(x0)
U36_gg(x0)
U37_gg(x0)
U38_gg(x0)
U29_ga(x0)
U31_ag(x0)
U32_ag(x0)
U33_ag(x0)
U34_ag(x0)
U35_ag(x0)
U36_ag(x0)
U37_ag(x0)
U38_ag(x0)
greater64_in_ag(x0)
U29_ag(x0)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MAX_VALUED3_IN_GGA(.(s(T18), T9), s(0)) → MAX_VALUED3_IN_GGA(T9, s(T18))
MAX_VALUED3_IN_GGA(.(s(s(T21)), T9), s(s(0))) → MAX_VALUED3_IN_GGA(T9, s(s(T21)))
MAX_VALUED3_IN_GGA(.(s(s(s(T24))), T9), s(s(s(0)))) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T27)))), T9), s(s(s(s(0))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T30))))), T9), s(s(s(s(s(0)))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T33)))))), T9), s(s(s(s(s(s(0))))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T36))))))), T9), s(s(s(s(s(s(s(0)))))))) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(s(T37))))))), T9), s(s(s(s(s(s(s(s(T38))))))))) → U9_GGA(T9, greater64_in_gg(T37, T38))
MAX_VALUED3_IN_GAA(.(s(T18), T9)) → MAX_VALUED3_IN_GGA(T9, s(T18))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(s(T43)))))), T44), s(s(s(s(s(s(s(T45)))))))) → U11_GGA(T44, higher_valued77_in_ag(T43))
MAX_VALUED3_IN_GAA(.(s(s(T21)), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(T21)))
MAX_VALUED3_IN_GGA(.(s(s(s(s(s(T52))))), T53), s(s(s(s(s(s(T54))))))) → U13_GGA(T53, T54, higher_valued82_in_gg(T54, T52))
MAX_VALUED3_IN_GGA(.(s(s(s(s(T59)))), T60), s(s(s(s(s(T61)))))) → U15_GGA(T60, T61, higher_valued87_in_gg(T61, T59))
MAX_VALUED3_IN_GGA(.(s(s(s(T66))), T67), s(s(s(s(T68))))) → U17_GGA(T67, T68, higher_valued92_in_gg(T68, T66))
MAX_VALUED3_IN_GGA(.(s(s(T73)), T74), s(s(s(T75)))) → U19_GGA(T74, T75, higher_valued97_in_gg(T75, T73))
MAX_VALUED3_IN_GGA(.(s(T80), T81), s(s(T82))) → U21_GGA(T81, T82, higher_valued102_in_gg(T82, T80))
MAX_VALUED3_IN_GGA(.(T87, T88), s(T89)) → U23_GGA(T88, T89, higher_valued107_in_gg(T89, T87))
MAX_VALUED3_IN_GGA(.(T94, T95), T96) → U25_GGA(T95, T96, higher_valued112_in_gg(T96, T94))
MAX_VALUED3_IN_GGA(.(T15, T9), 0) → MAX_VALUED3_IN_GGA(T9, T15)
U27_GGA(T102, T103, higher_valued118_out_gg) → MAX_VALUED3_IN_GGA(T102, T103)
MAX_VALUED3_IN_GAA(.(s(s(s(T24))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(T24))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T27)))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(T27)))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T30))))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(T30))))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T33)))))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(T33)))))))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T36))))))), T9)) → MAX_VALUED3_IN_GGA(T9, s(s(s(s(s(s(s(T36))))))))
U9_GAA(T9, greater64_out_ga(T38)) → MAX_VALUED3_IN_GAA(T9)
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(T43)))))), T44)) → U11_GAA(T44, higher_valued77_in_ag(T43))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(T52))))), T53)) → U13_GAA(T53, higher_valued82_in_ag(T52))
MAX_VALUED3_IN_GAA(.(s(s(s(s(T59)))), T60)) → U15_GAA(T60, higher_valued87_in_ag(T59))
MAX_VALUED3_IN_GAA(.(s(s(s(T66))), T67)) → U17_GAA(T67, higher_valued92_in_ag(T66))
MAX_VALUED3_IN_GAA(.(s(s(T73)), T74)) → U19_GAA(T74, higher_valued97_in_ag(T73))
MAX_VALUED3_IN_GAA(.(s(T80), T81)) → U21_GAA(T81, higher_valued102_in_ag(T80))
MAX_VALUED3_IN_GAA(.(T87, T88)) → U23_GAA(T88, higher_valued107_in_ag(T87))
MAX_VALUED3_IN_GAA(.(T94, T95)) → U25_GAA(T95, higher_valued112_in_ag(T94))
MAX_VALUED3_IN_GAA(.(T15, T9)) → MAX_VALUED3_IN_GGA(T9, T15)
MAX_VALUED3_IN_GAA(.(T101, T102)) → U27_GAA(T102, higher_valued118_in_ag(T101))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(MAX_VALUED3_IN_GAA(x₁)) = x₁   
POL(MAX_VALUED3_IN_GGA(x₁, x₂)) = x₁   
POL(U11_GAA(x₁, x₂)) = x₁   
POL(U11_GGA(x₁, x₂)) = x₁   
POL(U13_GAA(x₁, x₂)) = x₁   
POL(U13_GGA(x₁, x₂, x₃)) = x₁   
POL(U15_GAA(x₁, x₂)) = x₁   
POL(U15_GGA(x₁, x₂, x₃)) = x₁   
POL(U17_GAA(x₁, x₂)) = x₁   
POL(U17_GGA(x₁, x₂, x₃)) = x₁   
POL(U19_GAA(x₁, x₂)) = x₁   
POL(U19_GGA(x₁, x₂, x₃)) = x₁   
POL(U21_GAA(x₁, x₂)) = x₁   
POL(U21_GGA(x₁, x₂, x₃)) = x₁   
POL(U23_GAA(x₁, x₂)) = x₁   
POL(U23_GGA(x₁, x₂, x₃)) = x₁   
POL(U25_GAA(x₁, x₂)) = x₁   
POL(U25_GGA(x₁, x₂, x₃)) = x₁   
POL(U27_GAA(x₁, x₂)) = x₁   
POL(U27_GGA(x₁, x₂, x₃)) = 1 + x₁   
POL(U29_ag(x₁)) = 0   
POL(U29_ga(x₁)) = 0   
POL(U29_gg(x₁)) = 0   
POL(U30_ag(x₁)) = 0   
POL(U31_ag(x₁)) = 0   
POL(U31_gg(x₁)) = 0   
POL(U32_ag(x₁)) = 0   
POL(U32_gg(x₁)) = 0   
POL(U33_ag(x₁)) = 0   
POL(U33_gg(x₁)) = 0   
POL(U34_ag(x₁)) = 0   
POL(U34_gg(x₁)) = 1   
POL(U35_ag(x₁)) = 0   
POL(U35_gg(x₁)) = 0   
POL(U36_ag(x₁)) = 0   
POL(U36_gg(x₁)) = 0   
POL(U37_ag(x₁)) = 0   
POL(U37_gg(x₁)) = 0   
POL(U38_ag(x₁)) = 0   
POL(U38_gg(x₁)) = 1   
POL(U9_GAA(x₁, x₂)) = 1 + x₁   
POL(U9_GGA(x₁, x₂)) = x₁   
POL(greater64_in_ag(x₁)) = 1   
POL(greater64_in_ga(x₁)) = 0   
POL(greater64_in_gg(x₁, x₂)) = 0   
POL(greater64_out_ag) = 0   
POL(greater64_out_ga(x₁)) = 0   
POL(greater64_out_gg) = 0   
POL(higher_valued102_in_ag(x₁)) = 0   
POL(higher_valued102_in_gg(x₁, x₂)) = 0   
POL(higher_valued102_out_ag) = 0   
POL(higher_valued102_out_gg) = 0   
POL(higher_valued107_in_ag(x₁)) = x₁   
POL(higher_valued107_in_gg(x₁, x₂)) = 0   
POL(higher_valued107_out_ag) = 0   
POL(higher_valued107_out_gg) = 0   
POL(higher_valued112_in_ag(x₁)) = 0   
POL(higher_valued112_in_gg(x₁, x₂)) = 0   
POL(higher_valued112_out_ag) = 0   
POL(higher_valued112_out_gg) = 0   
POL(higher_valued118_in_ag(x₁)) = 0   
POL(higher_valued118_in_gg(x₁, x₂)) = 1   
POL(higher_valued118_out_ag) = 0   
POL(higher_valued118_out_gg) = 1   
POL(higher_valued77_in_ag(x₁)) = 0   
POL(higher_valued77_out_ag) = 0   
POL(higher_valued82_in_ag(x₁)) = 0   
POL(higher_valued82_in_gg(x₁, x₂)) = 0   
POL(higher_valued82_out_ag) = 0   
POL(higher_valued82_out_gg) = 0   
POL(higher_valued87_in_ag(x₁)) = 0   
POL(higher_valued87_in_gg(x₁, x₂)) = 1 + x₂   
POL(higher_valued87_out_ag) = 0   
POL(higher_valued87_out_gg) = 0   
POL(higher_valued92_in_ag(x₁)) = 0   
POL(higher_valued92_in_gg(x₁, x₂)) = 0   
POL(higher_valued92_out_ag) = 0   
POL(higher_valued92_out_gg) = 0   
POL(higher_valued97_in_ag(x₁)) = 0   
POL(higher_valued97_in_gg(x₁, x₂)) = 1 + x₂   
POL(higher_valued97_out_ag) = 0   
POL(higher_valued97_out_gg) = 0   
POL(s(x₁)) = 0   

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

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

U9_GGA(T9, greater64_out_gg) → MAX_VALUED3_IN_GAA(T9)
U11_GGA(T44, higher_valued77_out_ag) → MAX_VALUED3_IN_GAA(T44)
U13_GGA(T53, T54, higher_valued82_out_gg) → MAX_VALUED3_IN_GGA(T53, s(s(s(s(s(s(T54)))))))
U15_GGA(T60, T61, higher_valued87_out_gg) → MAX_VALUED3_IN_GGA(T60, s(s(s(s(s(T61))))))
U17_GGA(T67, T68, higher_valued92_out_gg) → MAX_VALUED3_IN_GGA(T67, s(s(s(s(T68)))))
U19_GGA(T74, T75, higher_valued97_out_gg) → MAX_VALUED3_IN_GGA(T74, s(s(s(T75))))
U21_GGA(T81, T82, higher_valued102_out_gg) → MAX_VALUED3_IN_GGA(T81, s(s(T82)))
U23_GGA(T88, T89, higher_valued107_out_gg) → MAX_VALUED3_IN_GGA(T88, s(T89))
U25_GGA(T95, T96, higher_valued112_out_gg) → MAX_VALUED3_IN_GGA(T95, T96)
MAX_VALUED3_IN_GGA(.(T101, T102), T103) → U27_GGA(T102, T103, higher_valued118_in_gg(T103, T101))
MAX_VALUED3_IN_GAA(.(s(s(s(s(s(s(s(T37))))))), T9)) → U9_GAA(T9, greater64_in_ga(T37))
U11_GAA(T44, higher_valued77_out_ag) → MAX_VALUED3_IN_GAA(T44)
U13_GAA(T53, higher_valued82_out_ag) → MAX_VALUED3_IN_GAA(T53)
U15_GAA(T60, higher_valued87_out_ag) → MAX_VALUED3_IN_GAA(T60)
U17_GAA(T67, higher_valued92_out_ag) → MAX_VALUED3_IN_GAA(T67)
U19_GAA(T74, higher_valued97_out_ag) → MAX_VALUED3_IN_GAA(T74)
U21_GAA(T81, higher_valued102_out_ag) → MAX_VALUED3_IN_GAA(T81)
U23_GAA(T88, higher_valued107_out_ag) → MAX_VALUED3_IN_GAA(T88)
U25_GAA(T95, higher_valued112_out_ag) → MAX_VALUED3_IN_GAA(T95)
U27_GAA(T102, higher_valued118_out_ag) → MAX_VALUED3_IN_GAA(T102)

The TRS R consists of the following rules:

greater64_in_gg(s(T40), 0) → greater64_out_gg
greater64_in_gg(s(T41), s(T42)) → U29_gg(greater64_in_gg(T41, T42))
higher_valued77_in_ag(T51) → U30_ag(greater64_in_ag(s(s(s(s(s(s(T51))))))))
higher_valued82_in_gg(T57, T58) → U31_gg(greater64_in_gg(s(s(s(s(s(s(s(T57))))))), s(s(s(s(s(T58)))))))
higher_valued87_in_gg(T64, T65) → U32_gg(greater64_in_gg(s(s(s(s(s(s(T64)))))), s(s(s(s(T65))))))
higher_valued92_in_gg(T71, T72) → U33_gg(greater64_in_gg(s(s(s(s(s(T71))))), s(s(s(T72)))))
higher_valued97_in_gg(T78, T79) → U34_gg(greater64_in_gg(s(s(s(s(T78)))), s(s(T79))))
higher_valued102_in_gg(T85, T86) → U35_gg(greater64_in_gg(s(s(s(T85))), s(T86)))
higher_valued107_in_gg(T92, T93) → U36_gg(greater64_in_gg(s(s(T92)), T93))
higher_valued112_in_gg(T99, T100) → U37_gg(greater64_in_gg(s(T99), T100))
higher_valued118_in_gg(T106, T107) → U38_gg(greater64_in_gg(s(T106), T107))
greater64_in_ga(s(T40)) → greater64_out_ga(0)
greater64_in_ga(s(T41)) → U29_ga(greater64_in_ga(T41))
higher_valued82_in_ag(T58) → U31_ag(greater64_in_ag(s(s(s(s(s(T58)))))))
higher_valued87_in_ag(T65) → U32_ag(greater64_in_ag(s(s(s(s(T65))))))
higher_valued92_in_ag(T72) → U33_ag(greater64_in_ag(s(s(s(T72)))))
higher_valued97_in_ag(T79) → U34_ag(greater64_in_ag(s(s(T79))))
higher_valued102_in_ag(T86) → U35_ag(greater64_in_ag(s(T86)))
higher_valued107_in_ag(T93) → U36_ag(greater64_in_ag(T93))
higher_valued112_in_ag(T100) → U37_ag(greater64_in_ag(T100))
higher_valued118_in_ag(T107) → U38_ag(greater64_in_ag(T107))
U29_gg(greater64_out_gg) → greater64_out_gg
U30_ag(greater64_out_ag) → higher_valued77_out_ag
U31_gg(greater64_out_gg) → higher_valued82_out_gg
U32_gg(greater64_out_gg) → higher_valued87_out_gg
U33_gg(greater64_out_gg) → higher_valued92_out_gg
U34_gg(greater64_out_gg) → higher_valued97_out_gg
U35_gg(greater64_out_gg) → higher_valued102_out_gg
U36_gg(greater64_out_gg) → higher_valued107_out_gg
U37_gg(greater64_out_gg) → higher_valued112_out_gg
U38_gg(greater64_out_gg) → higher_valued118_out_gg
U29_ga(greater64_out_ga(T42)) → greater64_out_ga(s(T42))
U31_ag(greater64_out_ag) → higher_valued82_out_ag
U32_ag(greater64_out_ag) → higher_valued87_out_ag
U33_ag(greater64_out_ag) → higher_valued92_out_ag
U34_ag(greater64_out_ag) → higher_valued97_out_ag
U35_ag(greater64_out_ag) → higher_valued102_out_ag
U36_ag(greater64_out_ag) → higher_valued107_out_ag
U37_ag(greater64_out_ag) → higher_valued112_out_ag
U38_ag(greater64_out_ag) → higher_valued118_out_ag
greater64_in_ag(s(T42)) → U29_ag(greater64_in_ag(T42))
greater64_in_ag(0) → greater64_out_ag
U29_ag(greater64_out_ag) → greater64_out_ag

The set Q consists of the following terms:

greater64_in_gg(x0, x1)
higher_valued77_in_ag(x0)
higher_valued82_in_gg(x0, x1)
higher_valued87_in_gg(x0, x1)
higher_valued92_in_gg(x0, x1)
higher_valued97_in_gg(x0, x1)
higher_valued102_in_gg(x0, x1)
higher_valued107_in_gg(x0, x1)
higher_valued112_in_gg(x0, x1)
higher_valued118_in_gg(x0, x1)
greater64_in_ga(x0)
higher_valued82_in_ag(x0)
higher_valued87_in_ag(x0)
higher_valued92_in_ag(x0)
higher_valued97_in_ag(x0)
higher_valued102_in_ag(x0)
higher_valued107_in_ag(x0)
higher_valued112_in_ag(x0)
higher_valued118_in_ag(x0)
U29_gg(x0)
U30_ag(x0)
U31_gg(x0)
U32_gg(x0)
U33_gg(x0)
U34_gg(x0)
U35_gg(x0)
U36_gg(x0)
U37_gg(x0)
U38_gg(x0)
U29_ga(x0)
U31_ag(x0)
U32_ag(x0)
U33_ag(x0)
U34_ag(x0)
U35_ag(x0)
U36_ag(x0)
U37_ag(x0)
U38_ag(x0)
greater64_in_ag(x0)
U29_ag(x0)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 20 less nodes.

(38) TRUE