Left Termination of the query pattern p_in_2(g, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

p(X, Y) :- ','(less(X, Y), ','(!, p(s(X), Y))).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

p(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1, T2), SCOPE: 1, CLAUSE: 0 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(less(T3, T4), ','(!_1, p(s(T3), T4))) | ?_1\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(less(T3, T4), ','(!_1, p(s(T3), T4))), SCOPE: 2, CLAUSE: 1 | ','(less(T3, T4), ','(!_1, p(s(T3), T4))), SCOPE: 2, CLAUSE: 2 | ?_1\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, p(s(0), s(T5))) | ','(less(0, s(T5)), ','(!_1, p(s(0), s(T5)))), SCOPE: 2, CLAUSE: 2 | ?_1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(less(T3, T4), ','(!_1, p(s(T3), T4))), SCOPE: 2, CLAUSE: 2 | ?_1\n\nT3 is ground\nT4 is ground\nX7 is free; \nless(T3, T4) !=? less(0, s(X7))", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(s(0), s(T5))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(s(0), s(T5)), SCOPE: 3, CLAUSE: 0 | ?_3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(less(s(0), s(T6)), ','(!_3, p(s(s(0)), s(T6)))) | ?_3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(less(s(0), s(T6)), ','(!_3, p(s(s(0)), s(T6)))), SCOPE: 4, CLAUSE: 1 | ','(less(s(0), s(T6)), ','(!_3, p(s(s(0)), s(T6)))), SCOPE: 4, CLAUSE: 2 | ?_3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(less(s(0), s(T6)), ','(!_3, p(s(s(0)), s(T6)))), SCOPE: 4, CLAUSE: 2 | ?_3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(less(0, T7), ','(!_3, p(s(s(0)), s(T7)))) | ?_3\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(less(0, T7), ','(!_3, p(s(s(0)), s(T7)))), SCOPE: 5, CLAUSE: 1 | ','(less(0, T7), ','(!_3, p(s(s(0)), s(T7)))), SCOPE: 5, CLAUSE: 2 | ?_3\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(!_3, p(s(s(0)), s(s(T8)))) | ','(less(0, s(T8)), ','(!_3, p(s(s(0)), s(s(T8))))), SCOPE: 5, CLAUSE: 2 | ?_3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(less(0, T7), ','(!_3, p(s(s(0)), s(T7)))), SCOPE: 5, CLAUSE: 2 | ?_3\n\nT7 is ground\nX18 is free; \nless(0, T7) !=? less(0, s(X18))", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(s(s(0)), s(s(T8)))\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(s(s(0)), s(s(T8))), SCOPE: 6, CLAUSE: 0 | ?_6\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(less(s(s(0)), s(s(T9))), ','(!_6, p(s(s(s(0))), s(s(T9))))) | ?_6\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(less(s(s(0)), s(s(T9))), ','(!_6, p(s(s(s(0))), s(s(T9))))), SCOPE: 7, CLAUSE: 1 | ','(less(s(s(0)), s(s(T9))), ','(!_6, p(s(s(s(0))), s(s(T9))))), SCOPE: 7, CLAUSE: 2 | ?_6\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(less(s(s(0)), s(s(T9))), ','(!_6, p(s(s(s(0))), s(s(T9))))), SCOPE: 7, CLAUSE: 2 | ?_6\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(less(s(0), s(T10)), ','(!_6, p(s(s(s(0))), s(s(T10))))) | ?_6\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(less(s(0), s(T10)), ','(!_6, p(s(s(s(0))), s(s(T10))))), SCOPE: 8, CLAUSE: 1 | ','(less(s(0), s(T10)), ','(!_6, p(s(s(s(0))), s(s(T10))))), SCOPE: 8, CLAUSE: 2 | ?_6\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(less(s(0), s(T10)), ','(!_6, p(s(s(s(0))), s(s(T10))))), SCOPE: 8, CLAUSE: 2 | ?_6\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(less(0, T11), ','(!_6, p(s(s(s(0))), s(s(T11))))) | ?_6\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(less(0, T11), ','(!_6, p(s(s(s(0))), s(s(T11))))), SCOPE: 9, CLAUSE: 1 | ','(less(0, T11), ','(!_6, p(s(s(s(0))), s(s(T11))))), SCOPE: 9, CLAUSE: 2 | ?_6\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(!_6, p(s(s(s(0))), s(s(s(T12))))) | ','(less(0, s(T12)), ','(!_6, p(s(s(s(0))), s(s(s(T12)))))), SCOPE: 9, CLAUSE: 2 | ?_6\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(less(0, T11), ','(!_6, p(s(s(s(0))), s(s(T11))))), SCOPE: 9, CLAUSE: 2 | ?_6\n\nT11 is ground\nX34 is free; \nless(0, T11) !=? less(0, s(X34))", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(s(s(s(0))), s(s(s(T12))))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(s(s(s(0))), s(s(s(T12)))), SCOPE: 10, CLAUSE: 0 | ?_10\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(less(s(s(s(0))), s(s(s(T13)))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T13)))))) | ?_10\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(less(s(s(s(0))), s(s(s(T13)))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T13)))))), SCOPE: 11, CLAUSE: 1 | ','(less(s(s(s(0))), s(s(s(T13)))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T13)))))), SCOPE: 11, CLAUSE: 2 | ?_10\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(less(s(s(s(0))), s(s(s(T13)))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T13)))))), SCOPE: 11, CLAUSE: 2 | ?_10\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(less(s(s(0)), s(s(T14))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T14)))))) | ?_10\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(less(s(s(0)), s(s(T14))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T14)))))), SCOPE: 12, CLAUSE: 1 | ','(less(s(s(0)), s(s(T14))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T14)))))), SCOPE: 12, CLAUSE: 2 | ?_10\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(less(s(s(0)), s(s(T14))), ','(!_10, p(s(s(s(s(0)))), s(s(s(T14)))))), SCOPE: 12, CLAUSE: 2 | ?_10\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(less(s(0), s(T15)), ','(!_10, p(s(s(s(s(0)))), s(s(s(T15)))))) | ?_10\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(less(s(0), s(T15)), ','(!_10, p(s(s(s(s(0)))), s(s(s(T15)))))), SCOPE: 13, CLAUSE: 1 | ','(less(s(0), s(T15)), ','(!_10, p(s(s(s(s(0)))), s(s(s(T15)))))), SCOPE: 13, CLAUSE: 2 | ?_10\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(less(s(0), s(T15)), ','(!_10, p(s(s(s(s(0)))), s(s(s(T15)))))), SCOPE: 13, CLAUSE: 2 | ?_10\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(less(0, T16), ','(!_10, p(s(s(s(s(0)))), s(s(s(T16)))))) | ?_10\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(less(0, T16), ','(!_10, p(s(s(s(s(0)))), s(s(s(T16)))))), SCOPE: 14, CLAUSE: 1 | ','(less(0, T16), ','(!_10, p(s(s(s(s(0)))), s(s(s(T16)))))), SCOPE: 14, CLAUSE: 2 | ?_10\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(!_10, p(s(s(s(s(0)))), s(s(s(s(T17)))))) | ','(less(0, s(T17)), ','(!_10, p(s(s(s(s(0)))), s(s(s(s(T17))))))), SCOPE: 14, CLAUSE: 2 | ?_10\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(less(0, T16), ','(!_10, p(s(s(s(s(0)))), s(s(s(T16)))))), SCOPE: 14, CLAUSE: 2 | ?_10\n\nT16 is ground\nX55 is free; \nless(0, T16) !=? less(0, s(X55))", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: p(s(s(s(s(0)))), s(s(s(s(T17)))))\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(s(s(s(s(0)))), s(s(s(s(T17))))), SCOPE: 15, CLAUSE: 0 | ?_15\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(less(s(s(s(s(0)))), s(s(s(s(T18))))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T18))))))) | ?_15\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(less(s(s(s(s(0)))), s(s(s(s(T18))))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T18))))))), SCOPE: 16, CLAUSE: 1 | ','(less(s(s(s(s(0)))), s(s(s(s(T18))))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T18))))))), SCOPE: 16, CLAUSE: 2 | ?_15\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(less(s(s(s(s(0)))), s(s(s(s(T18))))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T18))))))), SCOPE: 16, CLAUSE: 2 | ?_15\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(less(s(s(s(0))), s(s(s(T19)))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T19))))))) | ?_15\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(less(s(s(s(0))), s(s(s(T19)))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T19))))))), SCOPE: 17, CLAUSE: 1 | ','(less(s(s(s(0))), s(s(s(T19)))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T19))))))), SCOPE: 17, CLAUSE: 2 | ?_15\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(less(s(s(s(0))), s(s(s(T19)))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T19))))))), SCOPE: 17, CLAUSE: 2 | ?_15\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(less(s(s(0)), s(s(T20))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T20))))))) | ?_15\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(less(s(s(0)), s(s(T20))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T20))))))), SCOPE: 18, CLAUSE: 1 | ','(less(s(s(0)), s(s(T20))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T20))))))), SCOPE: 18, CLAUSE: 2 | ?_15\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(less(s(s(0)), s(s(T20))), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T20))))))), SCOPE: 18, CLAUSE: 2 | ?_15\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(less(s(0), s(T21)), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T21))))))) | ?_15\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(less(s(0), s(T21)), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T21))))))), SCOPE: 19, CLAUSE: 1 | ','(less(s(0), s(T21)), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T21))))))), SCOPE: 19, CLAUSE: 2 | ?_15\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(less(s(0), s(T21)), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T21))))))), SCOPE: 19, CLAUSE: 2 | ?_15\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(less(0, T22), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T22))))))) | ?_15\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(less(0, T22), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T22))))))), SCOPE: 20, CLAUSE: 1 | ','(less(0, T22), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T22))))))), SCOPE: 20, CLAUSE: 2 | ?_15\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(s(T23))))))) | ','(less(0, s(T23)), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(s(T23)))))))), SCOPE: 20, CLAUSE: 2 | ?_15\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(less(0, T22), ','(!_15, p(s(s(s(s(s(0))))), s(s(s(s(T22))))))), SCOPE: 20, CLAUSE: 2 | ?_15\n\nT22 is ground\nX81 is free; \nless(0, T22) !=? less(0, s(X81))", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: p(s(s(s(s(s(0))))), s(s(s(s(s(T23))))))\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: p(s(s(s(s(s(0))))), s(s(s(s(s(T23)))))), SCOPE: 21, CLAUSE: 0 | ?_21\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(less(s(s(s(s(s(0))))), s(s(s(s(s(T24)))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T24)))))))) | ?_21\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(less(s(s(s(s(s(0))))), s(s(s(s(s(T24)))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T24)))))))), SCOPE: 22, CLAUSE: 1 | ','(less(s(s(s(s(s(0))))), s(s(s(s(s(T24)))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T24)))))))), SCOPE: 22, CLAUSE: 2 | ?_21\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(less(s(s(s(s(s(0))))), s(s(s(s(s(T24)))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T24)))))))), SCOPE: 22, CLAUSE: 2 | ?_21\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(less(s(s(s(s(0)))), s(s(s(s(T25))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T25)))))))) | ?_21\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(less(s(s(s(s(0)))), s(s(s(s(T25))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T25)))))))), SCOPE: 23, CLAUSE: 1 | ','(less(s(s(s(s(0)))), s(s(s(s(T25))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T25)))))))), SCOPE: 23, CLAUSE: 2 | ?_21\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(less(s(s(s(s(0)))), s(s(s(s(T25))))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T25)))))))), SCOPE: 23, CLAUSE: 2 | ?_21\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(less(s(s(s(0))), s(s(s(T26)))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T26)))))))) | ?_21\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(less(s(s(s(0))), s(s(s(T26)))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T26)))))))), SCOPE: 24, CLAUSE: 1 | ','(less(s(s(s(0))), s(s(s(T26)))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T26)))))))), SCOPE: 24, CLAUSE: 2 | ?_21\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(less(s(s(s(0))), s(s(s(T26)))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T26)))))))), SCOPE: 24, CLAUSE: 2 | ?_21\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(less(s(s(0)), s(s(T27))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T27)))))))) | ?_21\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(less(s(s(0)), s(s(T27))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T27)))))))), SCOPE: 25, CLAUSE: 1 | ','(less(s(s(0)), s(s(T27))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T27)))))))), SCOPE: 25, CLAUSE: 2 | ?_21\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(less(s(s(0)), s(s(T27))), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T27)))))))), SCOPE: 25, CLAUSE: 2 | ?_21\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(less(s(0), s(T28)), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T28)))))))) | ?_21\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(less(s(0), s(T28)), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T28)))))))), SCOPE: 26, CLAUSE: 1 | ','(less(s(0), s(T28)), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T28)))))))), SCOPE: 26, CLAUSE: 2 | ?_21\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(less(s(0), s(T28)), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T28)))))))), SCOPE: 26, CLAUSE: 2 | ?_21\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(less(0, T29), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T29)))))))) | ?_21\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(less(0, T29), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T29)))))))), SCOPE: 27, CLAUSE: 1 | ','(less(0, T29), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T29)))))))), SCOPE: 27, CLAUSE: 2 | ?_21\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T30)))))))) | ','(less(0, s(T30)), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T30))))))))), SCOPE: 27, CLAUSE: 2 | ?_21\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(less(0, T29), ','(!_21, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(T29)))))))), SCOPE: 27, CLAUSE: 2 | ?_21\n\nT29 is ground\nX112 is free; \nless(0, T29) !=? less(0, s(X112))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T30)))))))\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T30))))))), SCOPE: 28, CLAUSE: 0 | ?_28\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(less(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T31))))))), ','(!_28, p(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(T31))))))))) | ?_28\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(less(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T31))))))), ','(!_28, p(s(T32), s(s(s(s(s(s(T31)))))))))\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: less(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T31)))))))\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(!_28, p(s(T32), s(s(s(s(s(s(T31))))))))\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: less(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T31))))))), SCOPE: 29, CLAUSE: 1 | less(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T31))))))), SCOPE: 29, CLAUSE: 2\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: less(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T31))))))), SCOPE: 29, CLAUSE: 2\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: less(s(s(s(s(s(0))))), s(s(s(s(s(T33))))))\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: less(s(s(s(s(s(0))))), s(s(s(s(s(T33)))))), SCOPE: 30, CLAUSE: 1 | less(s(s(s(s(s(0))))), s(s(s(s(s(T33)))))), SCOPE: 30, CLAUSE: 2\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: less(s(s(s(s(s(0))))), s(s(s(s(s(T33)))))), SCOPE: 30, CLAUSE: 2\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: less(s(s(s(s(0)))), s(s(s(s(T34)))))\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: less(s(s(s(s(0)))), s(s(s(s(T34))))), SCOPE: 31, CLAUSE: 1 | less(s(s(s(s(0)))), s(s(s(s(T34))))), SCOPE: 31, CLAUSE: 2\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: less(s(s(s(s(0)))), s(s(s(s(T34))))), SCOPE: 31, CLAUSE: 2\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: less(s(s(s(0))), s(s(s(T35))))\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: less(s(s(s(0))), s(s(s(T35)))), SCOPE: 32, CLAUSE: 1 | less(s(s(s(0))), s(s(s(T35)))), SCOPE: 32, CLAUSE: 2\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: less(s(s(s(0))), s(s(s(T35)))), SCOPE: 32, CLAUSE: 2\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: less(s(s(0)), s(s(T36)))\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: less(s(s(0)), s(s(T36))), SCOPE: 33, CLAUSE: 1 | less(s(s(0)), s(s(T36))), SCOPE: 33, CLAUSE: 2\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: less(s(s(0)), s(s(T36))), SCOPE: 33, CLAUSE: 2\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: less(s(0), s(T37))\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: less(s(0), s(T37)), SCOPE: 34, CLAUSE: 1 | less(s(0), s(T37)), SCOPE: 34, CLAUSE: 2\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: less(s(0), s(T37)), SCOPE: 34, CLAUSE: 2\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: less(0, T38)\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: less(0, T38), SCOPE: 35, CLAUSE: 1 | less(0, T38), SCOPE: 35, CLAUSE: 2\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: less(0, T38), SCOPE: 35, CLAUSE: 1\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: less(0, T38), SCOPE: 35, CLAUSE: 2\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: p(s(T32), s(s(s(s(s(s(T31)))))))\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: \n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(less(T40, T41), ','(!_1, p(s(s(T40)), s(T41)))) | ?_1\n\nT40 is ground\nT41 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: \n\nX7 is free\nX164 is free; \nX165 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: ','(less(T40, T41), ','(!_1, p(s(s(T40)), s(T41)))), SCOPE: 36, CLAUSE: 1 | ','(less(T40, T41), ','(!_1, p(s(s(T40)), s(T41)))), SCOPE: 36, CLAUSE: 2 | ?_1\n\nT40 is ground\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(!_1, p(s(s(0)), s(s(T42)))) | ','(less(0, s(T42)), ','(!_1, p(s(s(0)), s(s(T42))))), SCOPE: 36, CLAUSE: 2 | ?_1\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(less(T40, T41), ','(!_1, p(s(s(T40)), s(T41)))), SCOPE: 36, CLAUSE: 2 | ?_1\n\nT40 is ground\nT41 is ground\nX167 is free; \nless(T40, T41) !=? less(0, s(X167))", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: p(s(s(0)), s(s(T42)))\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(less(T43, T44), ','(!_1, p(s(s(s(T43))), s(s(T44))))) | ?_1\n\nT43 is ground\nT44 is ground\nX167 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\nX167 is free\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: ','(less(T43, T44), ','(!_1, p(s(s(s(T43))), s(s(T44))))), SCOPE: 37, CLAUSE: 1 | ','(less(T43, T44), ','(!_1, p(s(s(s(T43))), s(s(T44))))), SCOPE: 37, CLAUSE: 2 | ?_1\n\nT43 is ground\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: ','(!_1, p(s(s(s(0))), s(s(s(T45))))) | ','(less(0, s(T45)), ','(!_1, p(s(s(s(0))), s(s(s(T45)))))), SCOPE: 37, CLAUSE: 2 | ?_1\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: ','(less(T43, T44), ','(!_1, p(s(s(s(T43))), s(s(T44))))), SCOPE: 37, CLAUSE: 2 | ?_1\n\nT43 is ground\nT44 is ground\nX173 is free; \nless(T43, T44) !=? less(0, s(X173))", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: p(s(s(s(0))), s(s(s(T45))))\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: ','(less(T46, T47), ','(!_1, p(s(s(s(s(T46)))), s(s(s(T47)))))) | ?_1\n\nT46 is ground\nT47 is ground\nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\nX173 is free\nX176 is free; \nX177 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: ','(less(T46, T47), ','(!_1, p(s(s(s(s(T46)))), s(s(s(T47)))))), SCOPE: 38, CLAUSE: 1 | ','(less(T46, T47), ','(!_1, p(s(s(s(s(T46)))), s(s(s(T47)))))), SCOPE: 38, CLAUSE: 2 | ?_1\n\nT46 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: ','(!_1, p(s(s(s(s(0)))), s(s(s(s(T48)))))) | ','(less(0, s(T48)), ','(!_1, p(s(s(s(s(0)))), s(s(s(s(T48))))))), SCOPE: 38, CLAUSE: 2 | ?_1\n\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: ','(less(T46, T47), ','(!_1, p(s(s(s(s(T46)))), s(s(s(T47)))))), SCOPE: 38, CLAUSE: 2 | ?_1\n\nT46 is ground\nT47 is ground\nX179 is free; \nless(T46, T47) !=? less(0, s(X179))", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: p(s(s(s(s(0)))), s(s(s(s(T48)))))\n\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: ','(less(T49, T50), ','(!_1, p(s(s(s(s(s(T49))))), s(s(s(s(T50))))))) | ?_1\n\nT49 is ground\nT50 is ground\nX179 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: \n\nX179 is free\nX182 is free; \nX183 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: ','(less(T49, T50), ','(!_1, p(s(s(s(s(s(T49))))), s(s(s(s(T50))))))), SCOPE: 39, CLAUSE: 1 | ','(less(T49, T50), ','(!_1, p(s(s(s(s(s(T49))))), s(s(s(s(T50))))))), SCOPE: 39, CLAUSE: 2 | ?_1\n\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: ','(!_1, p(s(s(s(s(s(0))))), s(s(s(s(s(T51))))))) | ','(less(0, s(T51)), ','(!_1, p(s(s(s(s(s(0))))), s(s(s(s(s(T51)))))))), SCOPE: 39, CLAUSE: 2 | ?_1\n\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: ','(less(T49, T50), ','(!_1, p(s(s(s(s(s(T49))))), s(s(s(s(T50))))))), SCOPE: 39, CLAUSE: 2 | ?_1\n\nT49 is ground\nT50 is ground\nX185 is free; \nless(T49, T50) !=? less(0, s(X185))", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: p(s(s(s(s(s(0))))), s(s(s(s(s(T51))))))\n\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: ','(less(T52, T53), ','(!_1, p(s(s(s(s(s(s(T52)))))), s(s(s(s(s(T53)))))))) | ?_1\n\nT52 is ground\nT53 is ground\nX185 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\nX185 is free\nX188 is free; \nX189 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: ','(less(T52, T53), ','(!_1, p(s(s(s(s(s(s(T52)))))), s(s(s(s(s(T53)))))))), SCOPE: 40, CLAUSE: 1 | ','(less(T52, T53), ','(!_1, p(s(s(s(s(s(s(T52)))))), s(s(s(s(s(T53)))))))), SCOPE: 40, CLAUSE: 2 | ?_1\n\nT52 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: ','(!_1, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T54)))))))) | ','(less(0, s(T54)), ','(!_1, p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T54))))))))), SCOPE: 40, CLAUSE: 2 | ?_1\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: ','(less(T52, T53), ','(!_1, p(s(s(s(s(s(s(T52)))))), s(s(s(s(s(T53)))))))), SCOPE: 40, CLAUSE: 2 | ?_1\n\nT52 is ground\nT53 is ground\nX191 is free; \nless(T52, T53) !=? less(0, s(X191))", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: p(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(T54)))))))\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: ','(less(T55, T56), ','(!_1, p(s(s(s(s(s(s(s(T55))))))), s(s(s(s(s(s(T56))))))))) | ?_1\n\nT55 is ground\nT56 is ground\nX191 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: \n\nX191 is free\nX194 is free; \nX195 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: ','(less(T55, T56), ','(!_1, p(s(T57), s(s(s(s(s(s(T56)))))))))\n\nT55 is ground\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: less(T55, T56)\n\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: ','(!_1, p(s(T57), s(s(s(s(s(s(T56))))))))\n\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: less(T55, T56), SCOPE: 41, CLAUSE: 1 | less(T55, T56), SCOPE: 41, CLAUSE: 2\n\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: less(T55, T56), SCOPE: 41, CLAUSE: 1\n\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: less(T55, T56), SCOPE: 41, CLAUSE: 2\n\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: \n\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: less(T59, T60)\n\nT59 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: \n\nX201 is free\nX202 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: p(s(T57), s(s(s(s(s(s(T56)))))))\n\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 163 [arrowhead = none ];
163 -> 2;

164 [label="EVAL with clause\np(X4, X5) :- ','(less(X4, X5), ','(!, p(s(X4), X5))).\nand substitution\nT1 -> T3\nX4 -> T3\nT2 -> T4\nX5 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 164 [arrowhead = none ];
164 -> 3;

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

166 [label="EVAL with clause\nless(0, s(X7)).\nand substitution\nT3 -> 0\nX7 -> T5\nT4 -> s(T5)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 166 [arrowhead = none ];
166 -> 5;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 167 [arrowhead = none ];
167 -> 6;

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

169 [label="EVAL with clause\nless(s(X164), s(X165)) :- less(X164, X165).\nand substitution\nX164 -> T40\nT3 -> s(T40)\nX165 -> T41\nT4 -> s(T41)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 169 [arrowhead = none ];
169 -> 119;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 170 [arrowhead = none ];
170 -> 120;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 171 [arrowhead = none ];
171 -> 8;

172 [label="EVAL with clause\np(X10, X11) :- ','(less(X10, X11), ','(!, p(s(X10), X11))).\nand substitution\nX10 -> s(0)\nT5 -> T6\nX11 -> s(T6)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 172 [arrowhead = none ];
172 -> 9;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 173 [arrowhead = none ];
173 -> 10;

174 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 174 [arrowhead = none ];
174 -> 11;

175 [label="EVAL with clause\nless(s(X15), s(X16)) :- less(X15, X16).\nand substitution\nX15 -> 0\nT6 -> T7\nX16 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 175 [arrowhead = none ];
175 -> 12;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 176 [arrowhead = none ];
176 -> 13;

177 [label="EVAL with clause\nless(0, s(X18)).\nand substitution\nX18 -> T8\nT7 -> s(T8)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 177 [arrowhead = none ];
177 -> 14;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 178 [arrowhead = none ];
178 -> 15;

179 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 179 [arrowhead = none ];
179 -> 16;

180 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 180 [arrowhead = none ];
180 -> 118;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 181 [arrowhead = none ];
181 -> 17;

182 [label="EVAL with clause\np(X21, X22) :- ','(less(X21, X22), ','(!, p(s(X21), X22))).\nand substitution\nX21 -> s(s(0))\nT8 -> T9\nX22 -> s(s(T9))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 182 [arrowhead = none ];
182 -> 18;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 183 [arrowhead = none ];
183 -> 19;

184 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 184 [arrowhead = none ];
184 -> 20;

185 [label="EVAL with clause\nless(s(X26), s(X27)) :- less(X26, X27).\nand substitution\nX26 -> s(0)\nT9 -> T10\nX27 -> s(T10)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 185 [arrowhead = none ];
185 -> 21;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 186 [arrowhead = none ];
186 -> 22;

187 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
22 -> 187 [arrowhead = none ];
187 -> 23;

188 [label="EVAL with clause\nless(s(X31), s(X32)) :- less(X31, X32).\nand substitution\nX31 -> 0\nT10 -> T11\nX32 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 188 [arrowhead = none ];
188 -> 24;

189 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 189 [arrowhead = none ];
189 -> 25;

190 [label="EVAL with clause\nless(0, s(X34)).\nand substitution\nX34 -> T12\nT11 -> s(T12)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 190 [arrowhead = none ];
190 -> 26;

191 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 191 [arrowhead = none ];
191 -> 27;

192 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 192 [arrowhead = none ];
192 -> 28;

193 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 193 [arrowhead = none ];
193 -> 117;

194 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 194 [arrowhead = none ];
194 -> 29;

195 [label="EVAL with clause\np(X37, X38) :- ','(less(X37, X38), ','(!, p(s(X37), X38))).\nand substitution\nX37 -> s(s(s(0)))\nT12 -> T13\nX38 -> s(s(s(T13)))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 195 [arrowhead = none ];
195 -> 30;

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

197 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 197 [arrowhead = none ];
197 -> 32;

198 [label="EVAL with clause\nless(s(X42), s(X43)) :- less(X42, X43).\nand substitution\nX42 -> s(s(0))\nT13 -> T14\nX43 -> s(s(T14))", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 198 [arrowhead = none ];
198 -> 33;

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 199 [arrowhead = none ];
199 -> 34;

200 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
34 -> 200 [arrowhead = none ];
200 -> 35;

201 [label="EVAL with clause\nless(s(X47), s(X48)) :- less(X47, X48).\nand substitution\nX47 -> s(0)\nT14 -> T15\nX48 -> s(T15)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 201 [arrowhead = none ];
201 -> 36;

202 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 202 [arrowhead = none ];
202 -> 37;

203 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
37 -> 203 [arrowhead = none ];
203 -> 38;

204 [label="EVAL with clause\nless(s(X52), s(X53)) :- less(X52, X53).\nand substitution\nX52 -> 0\nT15 -> T16\nX53 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 204 [arrowhead = none ];
204 -> 39;

205 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 205 [arrowhead = none ];
205 -> 40;

206 [label="EVAL with clause\nless(0, s(X55)).\nand substitution\nX55 -> T17\nT16 -> s(T17)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 206 [arrowhead = none ];
206 -> 41;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 207 [arrowhead = none ];
207 -> 42;

208 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 208 [arrowhead = none ];
208 -> 43;

209 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
42 -> 209 [arrowhead = none ];
209 -> 116;

210 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 210 [arrowhead = none ];
210 -> 44;

211 [label="EVAL with clause\np(X58, X59) :- ','(less(X58, X59), ','(!, p(s(X58), X59))).\nand substitution\nX58 -> s(s(s(s(0))))\nT17 -> T18\nX59 -> s(s(s(s(T18))))", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 211 [arrowhead = none ];
211 -> 45;

212 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 212 [arrowhead = none ];
212 -> 46;

213 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
46 -> 213 [arrowhead = none ];
213 -> 47;

214 [label="EVAL with clause\nless(s(X63), s(X64)) :- less(X63, X64).\nand substitution\nX63 -> s(s(s(0)))\nT18 -> T19\nX64 -> s(s(s(T19)))", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 214 [arrowhead = none ];
214 -> 48;

215 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 215 [arrowhead = none ];
215 -> 49;

216 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 216 [arrowhead = none ];
216 -> 50;

217 [label="EVAL with clause\nless(s(X68), s(X69)) :- less(X68, X69).\nand substitution\nX68 -> s(s(0))\nT19 -> T20\nX69 -> s(s(T20))", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 217 [arrowhead = none ];
217 -> 51;

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 218 [arrowhead = none ];
218 -> 52;

219 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 219 [arrowhead = none ];
219 -> 53;

220 [label="EVAL with clause\nless(s(X73), s(X74)) :- less(X73, X74).\nand substitution\nX73 -> s(0)\nT20 -> T21\nX74 -> s(T21)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 220 [arrowhead = none ];
220 -> 54;

221 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 221 [arrowhead = none ];
221 -> 55;

222 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 222 [arrowhead = none ];
222 -> 56;

223 [label="EVAL with clause\nless(s(X78), s(X79)) :- less(X78, X79).\nand substitution\nX78 -> 0\nT21 -> T22\nX79 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 223 [arrowhead = none ];
223 -> 57;

224 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 224 [arrowhead = none ];
224 -> 58;

225 [label="EVAL with clause\nless(0, s(X81)).\nand substitution\nX81 -> T23\nT22 -> s(T23)", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 225 [arrowhead = none ];
225 -> 59;

226 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 226 [arrowhead = none ];
226 -> 60;

227 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 227 [arrowhead = none ];
227 -> 61;

228 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
60 -> 228 [arrowhead = none ];
228 -> 115;

229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 229 [arrowhead = none ];
229 -> 62;

230 [label="EVAL with clause\np(X84, X85) :- ','(less(X84, X85), ','(!, p(s(X84), X85))).\nand substitution\nX84 -> s(s(s(s(s(0)))))\nT23 -> T24\nX85 -> s(s(s(s(s(T24)))))", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 230 [arrowhead = none ];
230 -> 63;

231 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 231 [arrowhead = none ];
231 -> 64;

232 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
64 -> 232 [arrowhead = none ];
232 -> 65;

233 [label="EVAL with clause\nless(s(X89), s(X90)) :- less(X89, X90).\nand substitution\nX89 -> s(s(s(s(0))))\nT24 -> T25\nX90 -> s(s(s(s(T25))))", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 233 [arrowhead = none ];
233 -> 66;

234 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 234 [arrowhead = none ];
234 -> 67;

235 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
67 -> 235 [arrowhead = none ];
235 -> 68;

236 [label="EVAL with clause\nless(s(X94), s(X95)) :- less(X94, X95).\nand substitution\nX94 -> s(s(s(0)))\nT25 -> T26\nX95 -> s(s(s(T26)))", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 236 [arrowhead = none ];
236 -> 69;

237 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 237 [arrowhead = none ];
237 -> 70;

238 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 238 [arrowhead = none ];
238 -> 71;

239 [label="EVAL with clause\nless(s(X99), s(X100)) :- less(X99, X100).\nand substitution\nX99 -> s(s(0))\nT26 -> T27\nX100 -> s(s(T27))", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 239 [arrowhead = none ];
239 -> 72;

240 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 240 [arrowhead = none ];
240 -> 73;

241 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
73 -> 241 [arrowhead = none ];
241 -> 74;

242 [label="EVAL with clause\nless(s(X104), s(X105)) :- less(X104, X105).\nand substitution\nX104 -> s(0)\nT27 -> T28\nX105 -> s(T28)", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 242 [arrowhead = none ];
242 -> 75;

243 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 243 [arrowhead = none ];
243 -> 76;

244 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
76 -> 244 [arrowhead = none ];
244 -> 77;

245 [label="EVAL with clause\nless(s(X109), s(X110)) :- less(X109, X110).\nand substitution\nX109 -> 0\nT28 -> T29\nX110 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 245 [arrowhead = none ];
245 -> 78;

246 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
78 -> 246 [arrowhead = none ];
246 -> 79;

247 [label="EVAL with clause\nless(0, s(X112)).\nand substitution\nX112 -> T30\nT29 -> s(T30)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 247 [arrowhead = none ];
247 -> 80;

248 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 248 [arrowhead = none ];
248 -> 81;

249 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 249 [arrowhead = none ];
249 -> 82;

250 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
81 -> 250 [arrowhead = none ];
250 -> 114;

251 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 251 [arrowhead = none ];
251 -> 83;

252 [label="EVAL with clause\np(X115, X116) :- ','(less(X115, X116), ','(!, p(s(X115), X116))).\nand substitution\nX115 -> s(s(s(s(s(s(0))))))\nT30 -> T31\nX116 -> s(s(s(s(s(s(T31))))))", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 252 [arrowhead = none ];
252 -> 84;

253 [label="GENERALIZATION\nT32 <-- s(s(s(s(s(s(0))))))\n\nNew Knowledge:\nT32 is ground", fontsize=14, style = filled, fillcolor = violet];
84 -> 253 [arrowhead = none ];
253 -> 85;

254 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 254 [arrowhead = none ];
254 -> 86;

255 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
85 -> 255 [arrowhead = none ];
255 -> 87;

256 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
86 -> 256 [arrowhead = none ];
256 -> 88;

257 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 257 [arrowhead = none ];
257 -> 113;

258 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
88 -> 258 [arrowhead = none ];
258 -> 89;

259 [label="EVAL with clause\nless(s(X120), s(X121)) :- less(X120, X121).\nand substitution\nX120 -> s(s(s(s(s(0)))))\nT31 -> T33\nX121 -> s(s(s(s(s(T33)))))", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 259 [arrowhead = none ];
259 -> 90;

260 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
90 -> 260 [arrowhead = none ];
260 -> 91;

261 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
91 -> 261 [arrowhead = none ];
261 -> 92;

262 [label="EVAL with clause\nless(s(X125), s(X126)) :- less(X125, X126).\nand substitution\nX125 -> s(s(s(s(0))))\nT33 -> T34\nX126 -> s(s(s(s(T34))))", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 262 [arrowhead = none ];
262 -> 93;

263 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
93 -> 263 [arrowhead = none ];
263 -> 94;

264 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
94 -> 264 [arrowhead = none ];
264 -> 95;

265 [label="EVAL with clause\nless(s(X130), s(X131)) :- less(X130, X131).\nand substitution\nX130 -> s(s(s(0)))\nT34 -> T35\nX131 -> s(s(s(T35)))", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 265 [arrowhead = none ];
265 -> 96;

266 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
96 -> 266 [arrowhead = none ];
266 -> 97;

267 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
97 -> 267 [arrowhead = none ];
267 -> 98;

268 [label="EVAL with clause\nless(s(X135), s(X136)) :- less(X135, X136).\nand substitution\nX135 -> s(s(0))\nT35 -> T36\nX136 -> s(s(T36))", fontsize=14, style = filled, fillcolor = lightblue];
98 -> 268 [arrowhead = none ];
268 -> 99;

269 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 269 [arrowhead = none ];
269 -> 100;

270 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
100 -> 270 [arrowhead = none ];
270 -> 101;

271 [label="EVAL with clause\nless(s(X140), s(X141)) :- less(X140, X141).\nand substitution\nX140 -> s(0)\nT36 -> T37\nX141 -> s(T37)", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 271 [arrowhead = none ];
271 -> 102;

272 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 272 [arrowhead = none ];
272 -> 103;

273 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
103 -> 273 [arrowhead = none ];
273 -> 104;

274 [label="EVAL with clause\nless(s(X145), s(X146)) :- less(X145, X146).\nand substitution\nX145 -> 0\nT37 -> T38\nX146 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
104 -> 274 [arrowhead = none ];
274 -> 105;

275 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 275 [arrowhead = none ];
275 -> 106;

276 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 276 [arrowhead = none ];
276 -> 107;

277 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 277 [arrowhead = none ];
277 -> 108;

278 [label="EVAL with clause\nless(0, s(X149)).\nand substitution\nX149 -> T39\nT38 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 278 [arrowhead = none ];
278 -> 109;

279 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 279 [arrowhead = none ];
279 -> 110;

280 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
108 -> 280 [arrowhead = none ];
280 -> 112;

281 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 281 [arrowhead = none ];
281 -> 111;

282 [label="INSTANCE with matching:\nT1 -> s(T32)\nT2 -> s(s(s(s(s(s(T31))))))", fontsize=14, style = filled, fillcolor = lightgrey];
113 -> 282 [arrowhead = none , style=dashed];
282 -> 1[style=dashed];

283 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
119 -> 283 [arrowhead = none ];
283 -> 121;

284 [label="EVAL with clause\nless(0, s(X167)).\nand substitution\nT40 -> 0\nX167 -> T42\nT41 -> s(T42)", fontsize=14, style = filled, fillcolor = lightblue];
121 -> 284 [arrowhead = none ];
284 -> 122;

285 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
121 -> 285 [arrowhead = none ];
285 -> 123;

286 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 286 [arrowhead = none ];
286 -> 124;

287 [label="EVAL with clause\nless(s(X170), s(X171)) :- less(X170, X171).\nand substitution\nX170 -> T43\nT40 -> s(T43)\nX171 -> T44\nT41 -> s(T44)", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 287 [arrowhead = none ];
287 -> 125;

288 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 288 [arrowhead = none ];
288 -> 126;

289 [label="INSTANCE with matching:\nT8 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
124 -> 289 [arrowhead = none , style=dashed];
289 -> 16[style=dashed];

290 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 290 [arrowhead = none ];
290 -> 127;

291 [label="EVAL with clause\nless(0, s(X173)).\nand substitution\nT43 -> 0\nX173 -> T45\nT44 -> s(T45)", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 291 [arrowhead = none ];
291 -> 128;

292 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 292 [arrowhead = none ];
292 -> 129;

293 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 293 [arrowhead = none ];
293 -> 130;

294 [label="EVAL with clause\nless(s(X176), s(X177)) :- less(X176, X177).\nand substitution\nX176 -> T46\nT43 -> s(T46)\nX177 -> T47\nT44 -> s(T47)", fontsize=14, style = filled, fillcolor = lightblue];
129 -> 294 [arrowhead = none ];
294 -> 131;

295 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
129 -> 295 [arrowhead = none ];
295 -> 132;

296 [label="INSTANCE with matching:\nT12 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
130 -> 296 [arrowhead = none , style=dashed];
296 -> 28[style=dashed];

297 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
131 -> 297 [arrowhead = none ];
297 -> 133;

298 [label="EVAL with clause\nless(0, s(X179)).\nand substitution\nT46 -> 0\nX179 -> T48\nT47 -> s(T48)", fontsize=14, style = filled, fillcolor = lightblue];
133 -> 298 [arrowhead = none ];
298 -> 134;

299 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
133 -> 299 [arrowhead = none ];
299 -> 135;

300 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
134 -> 300 [arrowhead = none ];
300 -> 136;

301 [label="EVAL with clause\nless(s(X182), s(X183)) :- less(X182, X183).\nand substitution\nX182 -> T49\nT46 -> s(T49)\nX183 -> T50\nT47 -> s(T50)", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 301 [arrowhead = none ];
301 -> 137;

302 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 302 [arrowhead = none ];
302 -> 138;

303 [label="INSTANCE with matching:\nT17 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
136 -> 303 [arrowhead = none , style=dashed];
303 -> 43[style=dashed];

304 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
137 -> 304 [arrowhead = none ];
304 -> 139;

305 [label="EVAL with clause\nless(0, s(X185)).\nand substitution\nT49 -> 0\nX185 -> T51\nT50 -> s(T51)", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 305 [arrowhead = none ];
305 -> 140;

306 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 306 [arrowhead = none ];
306 -> 141;

307 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
140 -> 307 [arrowhead = none ];
307 -> 142;

308 [label="EVAL with clause\nless(s(X188), s(X189)) :- less(X188, X189).\nand substitution\nX188 -> T52\nT49 -> s(T52)\nX189 -> T53\nT50 -> s(T53)", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 308 [arrowhead = none ];
308 -> 143;

309 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 309 [arrowhead = none ];
309 -> 144;

310 [label="INSTANCE with matching:\nT23 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
142 -> 310 [arrowhead = none , style=dashed];
310 -> 61[style=dashed];

311 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
143 -> 311 [arrowhead = none ];
311 -> 145;

312 [label="EVAL with clause\nless(0, s(X191)).\nand substitution\nT52 -> 0\nX191 -> T54\nT53 -> s(T54)", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 312 [arrowhead = none ];
312 -> 146;

313 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 313 [arrowhead = none ];
313 -> 147;

314 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
146 -> 314 [arrowhead = none ];
314 -> 148;

315 [label="EVAL with clause\nless(s(X194), s(X195)) :- less(X194, X195).\nand substitution\nX194 -> T55\nT52 -> s(T55)\nX195 -> T56\nT53 -> s(T56)", fontsize=14, style = filled, fillcolor = lightblue];
147 -> 315 [arrowhead = none ];
315 -> 149;

316 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
147 -> 316 [arrowhead = none ];
316 -> 150;

317 [label="INSTANCE with matching:\nT30 -> T54", fontsize=14, style = filled, fillcolor = lightgrey];
148 -> 317 [arrowhead = none , style=dashed];
317 -> 82[style=dashed];

318 [label="GENERALIZATION\nT57 <-- s(s(s(s(s(s(T55))))))\n\nNew Knowledge:\nT57 is ground", fontsize=14, style = filled, fillcolor = violet];
149 -> 318 [arrowhead = none ];
318 -> 151;

319 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
151 -> 319 [arrowhead = none ];
319 -> 152;

320 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
151 -> 320 [arrowhead = none ];
320 -> 153;

321 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
152 -> 321 [arrowhead = none ];
321 -> 154;

322 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
153 -> 322 [arrowhead = none ];
322 -> 162;

323 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
154 -> 323 [arrowhead = none ];
323 -> 155;

324 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
154 -> 324 [arrowhead = none ];
324 -> 156;

325 [label="EVAL with clause\nless(0, s(X198)).\nand substitution\nT55 -> 0\nX198 -> T58\nT56 -> s(T58)", fontsize=14, style = filled, fillcolor = lightblue];
155 -> 325 [arrowhead = none ];
325 -> 157;

326 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
155 -> 326 [arrowhead = none ];
326 -> 158;

327 [label="EVAL with clause\nless(s(X201), s(X202)) :- less(X201, X202).\nand substitution\nX201 -> T59\nT55 -> s(T59)\nX202 -> T60\nT56 -> s(T60)", fontsize=14, style = filled, fillcolor = lightblue];
156 -> 327 [arrowhead = none ];
327 -> 160;

328 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
156 -> 328 [arrowhead = none ];
328 -> 161;

329 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
157 -> 329 [arrowhead = none ];
329 -> 159;

330 [label="INSTANCE with matching:\nT55 -> T59\nT56 -> T60", fontsize=14, style = filled, fillcolor = lightgrey];
160 -> 330 [arrowhead = none , style=dashed];
330 -> 152[style=dashed];

331 [label="INSTANCE with matching:\nT1 -> s(T57)\nT2 -> s(s(s(s(s(s(T56))))))", fontsize=14, style = filled, fillcolor = lightgrey];
162 -> 331 [arrowhead = none , style=dashed];
331 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

p1(0, s(s(s(s(s(s(T31))))))) :- less86(T31).
p1(0, s(s(s(s(s(s(T31))))))) :- ','(less86(T31), p1(s(T32), s(s(s(s(s(s(T31)))))))).
p1(s(0), s(s(T42))) :- p16(T42).
p1(s(s(0)), s(s(s(T45)))) :- p28(T45).
p1(s(s(s(0))), s(s(s(s(T48))))) :- p43(T48).
p1(s(s(s(s(0)))), s(s(s(s(s(T51)))))) :- p61(T51).
p1(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) :- p82(T54).
p1(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) :- less152(T55, T56).
p1(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) :- ','(less152(T55, T56), p1(s(T57), s(s(s(s(s(s(T56)))))))).
p16(s(s(s(s(T31))))) :- less86(T31).
p16(s(s(s(s(T31))))) :- ','(less86(T31), p1(s(T32), s(s(s(s(s(s(T31)))))))).
p28(s(s(s(T31)))) :- less86(T31).
p28(s(s(s(T31)))) :- ','(less86(T31), p1(s(T32), s(s(s(s(s(s(T31)))))))).
p43(s(s(T31))) :- less86(T31).
p43(s(s(T31))) :- ','(less86(T31), p1(s(T32), s(s(s(s(s(s(T31)))))))).
p61(s(T31)) :- less86(T31).
p61(s(T31)) :- ','(less86(T31), p1(s(T32), s(s(s(s(s(s(T31)))))))).
p82(T31) :- less86(T31).
p82(T31) :- ','(less86(T31), p1(s(T32), s(s(s(s(s(s(T31)))))))).
less152(0, s(T58)).
less152(s(T59), s(T60)) :- less152(T59, T60).
less86(s(T39)).

Queries:

p1(g,g).

(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:
p1_in: (b,b) (f,b)
p16_in: (b)
p28_in: (b)
p43_in: (b)
p61_in: (b)
p82_in: (b)
less152_in: (f,b) (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → U1_GG(T31, less86_in_g(T31))
P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_GG(T31, less86_out_g(T31)) → U2_GG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_GG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → U1_AG(T31, less86_in_g(T31))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_AG(T31, less86_out_g(T31)) → U2_AG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_AG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(0), s(s(T42))) → U3_AG(T42, p16_in_g(T42))
P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
P16_IN_G(s(s(s(s(T31))))) → LESS86_IN_G(T31)
U10_G(T31, less86_out_g(T31)) → U11_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → U4_AG(T45, p28_in_g(T45))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
P28_IN_G(s(s(s(T31)))) → LESS86_IN_G(T31)
U12_G(T31, less86_out_g(T31)) → U13_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → U5_AG(T48, p43_in_g(T48))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
P43_IN_G(s(s(T31))) → LESS86_IN_G(T31)
U14_G(T31, less86_out_g(T31)) → U15_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_AG(T51, p61_in_g(T51))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
P61_IN_G(s(T31)) → LESS86_IN_G(T31)
U16_G(T31, less86_out_g(T31)) → U17_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_AG(T54, p82_in_g(T54))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
P82_IN_G(T31) → LESS86_IN_G(T31)
U18_G(T31, less86_out_g(T31)) → U19_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_AG(T55, T56)
LESS152_IN_AG(s(T59), s(T60)) → U20_AG(T59, T60, less152_in_ag(T59, T60))
LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)
U8_AG(T55, T56, less152_out_ag(T55, T56)) → U9_AG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))
P1_IN_GG(s(0), s(s(T42))) → U3_GG(T42, p16_in_g(T42))
P1_IN_GG(s(0), s(s(T42))) → P16_IN_G(T42)
P1_IN_GG(s(s(0)), s(s(s(T45)))) → U4_GG(T45, p28_in_g(T45))
P1_IN_GG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → U5_GG(T48, p43_in_g(T48))
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_GG(T51, p61_in_g(T51))
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_GG(T54, p82_in_g(T54))
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_GG(T55, T56, less152_in_gg(T55, T56))
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_GG(T55, T56)
LESS152_IN_GG(s(T59), s(T60)) → U20_GG(T59, T60, less152_in_gg(T59, T60))
LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)
U8_GG(T55, T56, less152_out_gg(T55, T56)) → U9_GG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_GG(T55, T56, less152_out_gg(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
P1_IN_GG(x1, x2)  =  P1_IN_GG(x1, x2)
U1_GG(x1, x2)  =  U1_GG(x1, x2)
LESS86_IN_G(x1)  =  LESS86_IN_G(x1)
U2_GG(x1, x2)  =  U2_GG(x2)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
U1_AG(x1, x2)  =  U1_AG(x1, x2)
U2_AG(x1, x2)  =  U2_AG(x2)
U3_AG(x1, x2)  =  U3_AG(x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x2)
U4_AG(x1, x2)  =  U4_AG(x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x2)
U5_AG(x1, x2)  =  U5_AG(x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x2)
U6_AG(x1, x2)  =  U6_AG(x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x2)
U7_AG(x1, x2)  =  U7_AG(x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)
LESS152_IN_AG(x1, x2)  =  LESS152_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x3)
U9_AG(x1, x2, x3)  =  U9_AG(x1, x3)
U3_GG(x1, x2)  =  U3_GG(x2)
U4_GG(x1, x2)  =  U4_GG(x2)
U5_GG(x1, x2)  =  U5_GG(x2)
U6_GG(x1, x2)  =  U6_GG(x2)
U7_GG(x1, x2)  =  U7_GG(x2)
U8_GG(x1, x2, x3)  =  U8_GG(x2, x3)
LESS152_IN_GG(x1, x2)  =  LESS152_IN_GG(x1, x2)
U20_GG(x1, x2, x3)  =  U20_GG(x3)
U9_GG(x1, x2, x3)  =  U9_GG(x3)

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

(6) Obligation:

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

P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → U1_GG(T31, less86_in_g(T31))
P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_GG(T31, less86_out_g(T31)) → U2_GG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_GG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → U1_AG(T31, less86_in_g(T31))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_AG(T31, less86_out_g(T31)) → U2_AG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_AG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(0), s(s(T42))) → U3_AG(T42, p16_in_g(T42))
P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
P16_IN_G(s(s(s(s(T31))))) → LESS86_IN_G(T31)
U10_G(T31, less86_out_g(T31)) → U11_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → U4_AG(T45, p28_in_g(T45))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
P28_IN_G(s(s(s(T31)))) → LESS86_IN_G(T31)
U12_G(T31, less86_out_g(T31)) → U13_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → U5_AG(T48, p43_in_g(T48))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
P43_IN_G(s(s(T31))) → LESS86_IN_G(T31)
U14_G(T31, less86_out_g(T31)) → U15_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_AG(T51, p61_in_g(T51))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
P61_IN_G(s(T31)) → LESS86_IN_G(T31)
U16_G(T31, less86_out_g(T31)) → U17_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_AG(T54, p82_in_g(T54))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
P82_IN_G(T31) → LESS86_IN_G(T31)
U18_G(T31, less86_out_g(T31)) → U19_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_AG(T55, T56)
LESS152_IN_AG(s(T59), s(T60)) → U20_AG(T59, T60, less152_in_ag(T59, T60))
LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)
U8_AG(T55, T56, less152_out_ag(T55, T56)) → U9_AG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))
P1_IN_GG(s(0), s(s(T42))) → U3_GG(T42, p16_in_g(T42))
P1_IN_GG(s(0), s(s(T42))) → P16_IN_G(T42)
P1_IN_GG(s(s(0)), s(s(s(T45)))) → U4_GG(T45, p28_in_g(T45))
P1_IN_GG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → U5_GG(T48, p43_in_g(T48))
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_GG(T51, p61_in_g(T51))
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_GG(T54, p82_in_g(T54))
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_GG(T55, T56, less152_in_gg(T55, T56))
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_GG(T55, T56)
LESS152_IN_GG(s(T59), s(T60)) → U20_GG(T59, T60, less152_in_gg(T59, T60))
LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)
U8_GG(T55, T56, less152_out_gg(T55, T56)) → U9_GG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_GG(T55, T56, less152_out_gg(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
P1_IN_GG(x1, x2)  =  P1_IN_GG(x1, x2)
U1_GG(x1, x2)  =  U1_GG(x1, x2)
LESS86_IN_G(x1)  =  LESS86_IN_G(x1)
U2_GG(x1, x2)  =  U2_GG(x2)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
U1_AG(x1, x2)  =  U1_AG(x1, x2)
U2_AG(x1, x2)  =  U2_AG(x2)
U3_AG(x1, x2)  =  U3_AG(x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x2)
U4_AG(x1, x2)  =  U4_AG(x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x2)
U5_AG(x1, x2)  =  U5_AG(x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x2)
U6_AG(x1, x2)  =  U6_AG(x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x2)
U7_AG(x1, x2)  =  U7_AG(x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)
LESS152_IN_AG(x1, x2)  =  LESS152_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x3)
U9_AG(x1, x2, x3)  =  U9_AG(x1, x3)
U3_GG(x1, x2)  =  U3_GG(x2)
U4_GG(x1, x2)  =  U4_GG(x2)
U5_GG(x1, x2)  =  U5_GG(x2)
U6_GG(x1, x2)  =  U6_GG(x2)
U7_GG(x1, x2)  =  U7_GG(x2)
U8_GG(x1, x2, x3)  =  U8_GG(x2, x3)
LESS152_IN_GG(x1, x2)  =  LESS152_IN_GG(x1, x2)
U20_GG(x1, x2, x3)  =  U20_GG(x3)
U9_GG(x1, x2, x3)  =  U9_GG(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 41 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
LESS152_IN_GG(x1, x2)  =  LESS152_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)

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:

• LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)
  The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
LESS152_IN_AG(x1, x2)  =  LESS152_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:

LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS152_IN_AG(x1, x2)  =  LESS152_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:

LESS152_IN_AG(s(T60)) → LESS152_IN_AG(T60)

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:

• LESS152_IN_AG(s(T60)) → LESS152_IN_AG(T60)
  The graph contains the following edges 1 > 1

(22) TRUE

(23) Obligation:

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

P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
p1_out_gg(x1, x2)  =  p1_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1)
U2_ag(x1, x2)  =  U2_ag(x2)
U3_ag(x1, x2)  =  U3_ag(x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g
U11_g(x1, x2)  =  U11_g(x2)
U4_ag(x1, x2)  =  U4_ag(x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g
U13_g(x1, x2)  =  U13_g(x2)
U5_ag(x1, x2)  =  U5_ag(x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g
U15_g(x1, x2)  =  U15_g(x2)
U6_ag(x1, x2)  =  U6_ag(x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g
U17_g(x1, x2)  =  U17_g(x2)
U7_ag(x1, x2)  =  U7_ag(x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g
U19_g(x1, x2)  =  U19_g(x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x3)
U3_gg(x1, x2)  =  U3_gg(x2)
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2)  =  U5_gg(x2)
U6_gg(x1, x2)  =  U6_gg(x2)
U7_gg(x1, x2)  =  U7_gg(x2)
U8_gg(x1, x2, x3)  =  U8_gg(x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg
U20_gg(x1, x2, x3)  =  U20_gg(x3)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)

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:

P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1)
U20_ag(x1, x2, x3)  =  U20_ag(x3)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(28) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(T42))) → P16_IN_G(T42) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

(29) Obligation:

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

P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)

(31) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(32) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)

(33) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(34) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)

(35) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(36) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)

(37) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(38) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P82_IN_G(T31) → U18_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)

(39) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g
less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(40) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(41) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0)

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

(42) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

less86_in_g(x0)

(43) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(44) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(T42))) → P16_IN_G(T42) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

(45) Obligation:

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

U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(46) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56)) at position [1] we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))

(47) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(48) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(T42))) → P16_IN_G(T42) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

(49) Obligation:

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

U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(50) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U10_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(51) Obligation:

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

P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(52) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))

(53) Obligation:

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

U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(54) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(55) Obligation:

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

P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(56) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))

(57) Obligation:

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

U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(58) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(59) Obligation:

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

P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(60) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))

(61) Obligation:

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

U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(62) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(63) Obligation:

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

P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(64) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U18_G(T31, less86_out_g) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(65) Obligation:

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

P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(66) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U8_AG(T56, less152_out_ag(T55)) → P1_IN_AG(s(s(s(s(s(s(T56))))))) we obtained the following new rules [LPAR04]:

U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(67) Obligation:

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

P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(68) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))

(69) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(70) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0))))) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

(71) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(72) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0)))) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))

(73) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(74) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0))) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))

(75) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(76) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0)) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))

(77) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g)
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g)
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g)
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g)
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g)
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(less152_in_ag(x0)))
U10_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0)
less152_in_ag(s(T60)) → U20_ag(less152_in_ag(T60))
U20_ag(less152_out_ag(T59)) → less152_out_ag(s(T59))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0)

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

(78) NonTerminationProof (EQUIVALENT transformation)

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

s = U8_AG(s(z0), less152_out_ag(0)) evaluates to t =U8_AG(s(z0), less152_out_ag(0))

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

U8_AG(s(z0), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
with rule U8_AG(s(z0'), less152_out_ag(0)) → P1_IN_AG(s(s(s(s(s(s(s(z0')))))))) at position [] and matcher [z0' / z0]

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → U8_AG(s(z0), less152_out_ag(0))
with rule P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0))

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


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

(79) FALSE

(80) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b,b) (f,b)
p16_in: (b)
p28_in: (b)
p43_in: (b)
p61_in: (b)
p82_in: (b)
less152_in: (f,b) (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(81) Obligation:

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

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)

(82) 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:

P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → U1_GG(T31, less86_in_g(T31))
P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_GG(T31, less86_out_g(T31)) → U2_GG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_GG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → U1_AG(T31, less86_in_g(T31))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_AG(T31, less86_out_g(T31)) → U2_AG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_AG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(0), s(s(T42))) → U3_AG(T42, p16_in_g(T42))
P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
P16_IN_G(s(s(s(s(T31))))) → LESS86_IN_G(T31)
U10_G(T31, less86_out_g(T31)) → U11_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → U4_AG(T45, p28_in_g(T45))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
P28_IN_G(s(s(s(T31)))) → LESS86_IN_G(T31)
U12_G(T31, less86_out_g(T31)) → U13_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → U5_AG(T48, p43_in_g(T48))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
P43_IN_G(s(s(T31))) → LESS86_IN_G(T31)
U14_G(T31, less86_out_g(T31)) → U15_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_AG(T51, p61_in_g(T51))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
P61_IN_G(s(T31)) → LESS86_IN_G(T31)
U16_G(T31, less86_out_g(T31)) → U17_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_AG(T54, p82_in_g(T54))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
P82_IN_G(T31) → LESS86_IN_G(T31)
U18_G(T31, less86_out_g(T31)) → U19_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_AG(T55, T56)
LESS152_IN_AG(s(T59), s(T60)) → U20_AG(T59, T60, less152_in_ag(T59, T60))
LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)
U8_AG(T55, T56, less152_out_ag(T55, T56)) → U9_AG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))
P1_IN_GG(s(0), s(s(T42))) → U3_GG(T42, p16_in_g(T42))
P1_IN_GG(s(0), s(s(T42))) → P16_IN_G(T42)
P1_IN_GG(s(s(0)), s(s(s(T45)))) → U4_GG(T45, p28_in_g(T45))
P1_IN_GG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → U5_GG(T48, p43_in_g(T48))
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_GG(T51, p61_in_g(T51))
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_GG(T54, p82_in_g(T54))
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_GG(T55, T56, less152_in_gg(T55, T56))
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_GG(T55, T56)
LESS152_IN_GG(s(T59), s(T60)) → U20_GG(T59, T60, less152_in_gg(T59, T60))
LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)
U8_GG(T55, T56, less152_out_gg(T55, T56)) → U9_GG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_GG(T55, T56, less152_out_gg(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
P1_IN_GG(x1, x2)  =  P1_IN_GG(x1, x2)
U1_GG(x1, x2)  =  U1_GG(x1, x2)
LESS86_IN_G(x1)  =  LESS86_IN_G(x1)
U2_GG(x1, x2)  =  U2_GG(x1, x2)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
U1_AG(x1, x2)  =  U1_AG(x1, x2)
U2_AG(x1, x2)  =  U2_AG(x1, x2)
U3_AG(x1, x2)  =  U3_AG(x1, x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
U4_AG(x1, x2)  =  U4_AG(x1, x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)
U5_AG(x1, x2)  =  U5_AG(x1, x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U6_AG(x1, x2)  =  U6_AG(x1, x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U7_AG(x1, x2)  =  U7_AG(x1, x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)
LESS152_IN_AG(x1, x2)  =  LESS152_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x2, x3)
U9_AG(x1, x2, x3)  =  U9_AG(x1, x2, x3)
U3_GG(x1, x2)  =  U3_GG(x1, x2)
U4_GG(x1, x2)  =  U4_GG(x1, x2)
U5_GG(x1, x2)  =  U5_GG(x1, x2)
U6_GG(x1, x2)  =  U6_GG(x1, x2)
U7_GG(x1, x2)  =  U7_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
LESS152_IN_GG(x1, x2)  =  LESS152_IN_GG(x1, x2)
U20_GG(x1, x2, x3)  =  U20_GG(x1, x2, x3)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)

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

(83) Obligation:

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

P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → U1_GG(T31, less86_in_g(T31))
P1_IN_GG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_GG(T31, less86_out_g(T31)) → U2_GG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_GG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → U1_AG(T31, less86_in_g(T31))
P1_IN_AG(0, s(s(s(s(s(s(T31))))))) → LESS86_IN_G(T31)
U1_AG(T31, less86_out_g(T31)) → U2_AG(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U1_AG(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(0), s(s(T42))) → U3_AG(T42, p16_in_g(T42))
P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
P16_IN_G(s(s(s(s(T31))))) → LESS86_IN_G(T31)
U10_G(T31, less86_out_g(T31)) → U11_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → U4_AG(T45, p28_in_g(T45))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
P28_IN_G(s(s(s(T31)))) → LESS86_IN_G(T31)
U12_G(T31, less86_out_g(T31)) → U13_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → U5_AG(T48, p43_in_g(T48))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
P43_IN_G(s(s(T31))) → LESS86_IN_G(T31)
U14_G(T31, less86_out_g(T31)) → U15_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_AG(T51, p61_in_g(T51))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
P61_IN_G(s(T31)) → LESS86_IN_G(T31)
U16_G(T31, less86_out_g(T31)) → U17_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_AG(T54, p82_in_g(T54))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
P82_IN_G(T31) → LESS86_IN_G(T31)
U18_G(T31, less86_out_g(T31)) → U19_G(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_AG(T55, T56)
LESS152_IN_AG(s(T59), s(T60)) → U20_AG(T59, T60, less152_in_ag(T59, T60))
LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)
U8_AG(T55, T56, less152_out_ag(T55, T56)) → U9_AG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))
P1_IN_GG(s(0), s(s(T42))) → U3_GG(T42, p16_in_g(T42))
P1_IN_GG(s(0), s(s(T42))) → P16_IN_G(T42)
P1_IN_GG(s(s(0)), s(s(s(T45)))) → U4_GG(T45, p28_in_g(T45))
P1_IN_GG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → U5_GG(T48, p43_in_g(T48))
P1_IN_GG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_GG(T51, p61_in_g(T51))
P1_IN_GG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_GG(T54, p82_in_g(T54))
P1_IN_GG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_GG(T55, T56, less152_in_gg(T55, T56))
P1_IN_GG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → LESS152_IN_GG(T55, T56)
LESS152_IN_GG(s(T59), s(T60)) → U20_GG(T59, T60, less152_in_gg(T59, T60))
LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)
U8_GG(T55, T56, less152_out_gg(T55, T56)) → U9_GG(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U8_GG(T55, T56, less152_out_gg(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
P1_IN_GG(x1, x2)  =  P1_IN_GG(x1, x2)
U1_GG(x1, x2)  =  U1_GG(x1, x2)
LESS86_IN_G(x1)  =  LESS86_IN_G(x1)
U2_GG(x1, x2)  =  U2_GG(x1, x2)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
U1_AG(x1, x2)  =  U1_AG(x1, x2)
U2_AG(x1, x2)  =  U2_AG(x1, x2)
U3_AG(x1, x2)  =  U3_AG(x1, x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
U4_AG(x1, x2)  =  U4_AG(x1, x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)
U5_AG(x1, x2)  =  U5_AG(x1, x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U6_AG(x1, x2)  =  U6_AG(x1, x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U7_AG(x1, x2)  =  U7_AG(x1, x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)
LESS152_IN_AG(x1, x2)  =  LESS152_IN_AG(x2)
U20_AG(x1, x2, x3)  =  U20_AG(x2, x3)
U9_AG(x1, x2, x3)  =  U9_AG(x1, x2, x3)
U3_GG(x1, x2)  =  U3_GG(x1, x2)
U4_GG(x1, x2)  =  U4_GG(x1, x2)
U5_GG(x1, x2)  =  U5_GG(x1, x2)
U6_GG(x1, x2)  =  U6_GG(x1, x2)
U7_GG(x1, x2)  =  U7_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
LESS152_IN_GG(x1, x2)  =  LESS152_IN_GG(x1, x2)
U20_GG(x1, x2, x3)  =  U20_GG(x1, x2, x3)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)

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

(84) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 41 less nodes.

(85) Complex Obligation (AND)

(86) Obligation:

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

LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
LESS152_IN_GG(x1, x2)  =  LESS152_IN_GG(x1, x2)

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

(87) UsableRulesProof (EQUIVALENT transformation)

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

(88) Obligation:

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

LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)

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

(89) PiDPToQDPProof (EQUIVALENT transformation)

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

(90) Obligation:

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

LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)

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

(91) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS152_IN_GG(s(T59), s(T60)) → LESS152_IN_GG(T59, T60)
  The graph contains the following edges 1 > 1, 2 > 2

(92) TRUE

(93) Obligation:

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

LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
LESS152_IN_AG(x1, x2)  =  LESS152_IN_AG(x2)

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

(94) UsableRulesProof (EQUIVALENT transformation)

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

(95) Obligation:

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

LESS152_IN_AG(s(T59), s(T60)) → LESS152_IN_AG(T59, T60)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS152_IN_AG(x1, x2)  =  LESS152_IN_AG(x2)

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

(96) PiDPToQDPProof (SOUND transformation)

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

(97) Obligation:

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

LESS152_IN_AG(s(T60)) → LESS152_IN_AG(T60)

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

(98) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS152_IN_AG(s(T60)) → LESS152_IN_AG(T60)
  The graph contains the following edges 1 > 1

(99) TRUE

(100) Obligation:

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

P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

p1_in_gg(0, s(s(s(s(s(s(T31))))))) → U1_gg(T31, less86_in_g(T31))
less86_in_g(s(T39)) → less86_out_g(s(T39))
U1_gg(T31, less86_out_g(T31)) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
U1_gg(T31, less86_out_g(T31)) → U2_gg(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(0, s(s(s(s(s(s(T31))))))) → U1_ag(T31, less86_in_g(T31))
U1_ag(T31, less86_out_g(T31)) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U1_ag(T31, less86_out_g(T31)) → U2_ag(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(0), s(s(T42))) → U3_ag(T42, p16_in_g(T42))
p16_in_g(s(s(s(s(T31))))) → U10_g(T31, less86_in_g(T31))
U10_g(T31, less86_out_g(T31)) → p16_out_g(s(s(s(s(T31)))))
U10_g(T31, less86_out_g(T31)) → U11_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(0)), s(s(s(T45)))) → U4_ag(T45, p28_in_g(T45))
p28_in_g(s(s(s(T31)))) → U12_g(T31, less86_in_g(T31))
U12_g(T31, less86_out_g(T31)) → p28_out_g(s(s(s(T31))))
U12_g(T31, less86_out_g(T31)) → U13_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(0))), s(s(s(s(T48))))) → U5_ag(T48, p43_in_g(T48))
p43_in_g(s(s(T31))) → U14_g(T31, less86_in_g(T31))
U14_g(T31, less86_out_g(T31)) → p43_out_g(s(s(T31)))
U14_g(T31, less86_out_g(T31)) → U15_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_ag(T51, p61_in_g(T51))
p61_in_g(s(T31)) → U16_g(T31, less86_in_g(T31))
U16_g(T31, less86_out_g(T31)) → p61_out_g(s(T31))
U16_g(T31, less86_out_g(T31)) → U17_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_ag(T54, p82_in_g(T54))
p82_in_g(T31) → U18_g(T31, less86_in_g(T31))
U18_g(T31, less86_out_g(T31)) → p82_out_g(T31)
U18_g(T31, less86_out_g(T31)) → U19_g(T31, p1_in_ag(s(T32), s(s(s(s(s(s(T31))))))))
p1_in_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_ag(T55, T56, less152_in_ag(T55, T56))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_ag(T55, T56, less152_out_ag(T55, T56)) → U9_ag(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_ag(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_ag(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U19_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p82_out_g(T31)
U7_ag(T54, p82_out_g(T54)) → p1_out_ag(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
U17_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p61_out_g(s(T31))
U6_ag(T51, p61_out_g(T51)) → p1_out_ag(s(s(s(s(0)))), s(s(s(s(s(T51))))))
U15_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p43_out_g(s(s(T31)))
U5_ag(T48, p43_out_g(T48)) → p1_out_ag(s(s(s(0))), s(s(s(s(T48)))))
U13_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p28_out_g(s(s(s(T31))))
U4_ag(T45, p28_out_g(T45)) → p1_out_ag(s(s(0)), s(s(s(T45))))
U11_g(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p16_out_g(s(s(s(s(T31)))))
U3_ag(T42, p16_out_g(T42)) → p1_out_ag(s(0), s(s(T42)))
U2_ag(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_ag(0, s(s(s(s(s(s(T31)))))))
U2_gg(T31, p1_out_ag(s(T32), s(s(s(s(s(s(T31)))))))) → p1_out_gg(0, s(s(s(s(s(s(T31)))))))
p1_in_gg(s(0), s(s(T42))) → U3_gg(T42, p16_in_g(T42))
U3_gg(T42, p16_out_g(T42)) → p1_out_gg(s(0), s(s(T42)))
p1_in_gg(s(s(0)), s(s(s(T45)))) → U4_gg(T45, p28_in_g(T45))
U4_gg(T45, p28_out_g(T45)) → p1_out_gg(s(s(0)), s(s(s(T45))))
p1_in_gg(s(s(s(0))), s(s(s(s(T48))))) → U5_gg(T48, p43_in_g(T48))
U5_gg(T48, p43_out_g(T48)) → p1_out_gg(s(s(s(0))), s(s(s(s(T48)))))
p1_in_gg(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → U6_gg(T51, p61_in_g(T51))
U6_gg(T51, p61_out_g(T51)) → p1_out_gg(s(s(s(s(0)))), s(s(s(s(s(T51))))))
p1_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → U7_gg(T54, p82_in_g(T54))
U7_gg(T54, p82_out_g(T54)) → p1_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T54)))))))
p1_in_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_gg(T55, T56, less152_in_gg(T55, T56))
less152_in_gg(0, s(T58)) → less152_out_gg(0, s(T58))
less152_in_gg(s(T59), s(T60)) → U20_gg(T59, T60, less152_in_gg(T59, T60))
U20_gg(T59, T60, less152_out_gg(T59, T60)) → less152_out_gg(s(T59), s(T60))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))
U8_gg(T55, T56, less152_out_gg(T55, T56)) → U9_gg(T55, T56, p1_in_ag(s(T57), s(s(s(s(s(s(T56))))))))
U9_gg(T55, T56, p1_out_ag(s(T57), s(s(s(s(s(s(T56)))))))) → p1_out_gg(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56)))))))

The argument filtering Pi contains the following mapping:
p1_in_gg(x1, x2)  =  p1_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
U1_gg(x1, x2)  =  U1_gg(x1, x2)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
p1_out_gg(x1, x2)  =  p1_out_gg(x1, x2)
U2_gg(x1, x2)  =  U2_gg(x1, x2)
p1_in_ag(x1, x2)  =  p1_in_ag(x2)
U1_ag(x1, x2)  =  U1_ag(x1, x2)
p1_out_ag(x1, x2)  =  p1_out_ag(x1, x2)
U2_ag(x1, x2)  =  U2_ag(x1, x2)
U3_ag(x1, x2)  =  U3_ag(x1, x2)
p16_in_g(x1)  =  p16_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
p16_out_g(x1)  =  p16_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
U4_ag(x1, x2)  =  U4_ag(x1, x2)
p28_in_g(x1)  =  p28_in_g(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
p28_out_g(x1)  =  p28_out_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
U5_ag(x1, x2)  =  U5_ag(x1, x2)
p43_in_g(x1)  =  p43_in_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
p43_out_g(x1)  =  p43_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U6_ag(x1, x2)  =  U6_ag(x1, x2)
p61_in_g(x1)  =  p61_in_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
p61_out_g(x1)  =  p61_out_g(x1)
U17_g(x1, x2)  =  U17_g(x1, x2)
U7_ag(x1, x2)  =  U7_ag(x1, x2)
p82_in_g(x1)  =  p82_in_g(x1)
U18_g(x1, x2)  =  U18_g(x1, x2)
p82_out_g(x1)  =  p82_out_g(x1)
U19_g(x1, x2)  =  U19_g(x1, x2)
U8_ag(x1, x2, x3)  =  U8_ag(x2, x3)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
U9_ag(x1, x2, x3)  =  U9_ag(x1, x2, x3)
U3_gg(x1, x2)  =  U3_gg(x1, x2)
U4_gg(x1, x2)  =  U4_gg(x1, x2)
U5_gg(x1, x2)  =  U5_gg(x1, x2)
U6_gg(x1, x2)  =  U6_gg(x1, x2)
U7_gg(x1, x2)  =  U7_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
less152_in_gg(x1, x2)  =  less152_in_gg(x1, x2)
less152_out_gg(x1, x2)  =  less152_out_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)

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

(101) UsableRulesProof (EQUIVALENT transformation)

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

(102) Obligation:

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

P1_IN_AG(s(0), s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(0)), s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(0))), s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(0)))), s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(0))))), s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(T32), s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T55)))))), s(s(s(s(s(s(T56))))))) → U8_AG(T55, T56, less152_in_ag(T55, T56))
U8_AG(T55, T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(T57), s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(0, s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T59), s(T60)) → U20_ag(T59, T60, less152_in_ag(T59, T60))
U20_ag(T59, T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
less86_in_g(x1)  =  less86_in_g(x1)
less86_out_g(x1)  =  less86_out_g(x1)
less152_in_ag(x1, x2)  =  less152_in_ag(x2)
less152_out_ag(x1, x2)  =  less152_out_ag(x1, x2)
U20_ag(x1, x2, x3)  =  U20_ag(x2, x3)
P1_IN_AG(x1, x2)  =  P1_IN_AG(x2)
P16_IN_G(x1)  =  P16_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
P28_IN_G(x1)  =  P28_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
P43_IN_G(x1)  =  P43_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
P61_IN_G(x1)  =  P61_IN_G(x1)
U16_G(x1, x2)  =  U16_G(x1, x2)
P82_IN_G(x1)  =  P82_IN_G(x1)
U18_G(x1, x2)  =  U18_G(x1, x2)
U8_AG(x1, x2, x3)  =  U8_AG(x2, x3)

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

(103) PiDPToQDPProof (SOUND transformation)

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

(104) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(105) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(T42))) → P16_IN_G(T42) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

(106) Obligation:

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

P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31))
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(107) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P16_IN_G(s(s(s(s(T31))))) → U10_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))

(108) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31))
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(109) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P28_IN_G(s(s(s(T31)))) → U12_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))

(110) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31))
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(111) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P43_IN_G(s(s(T31))) → U14_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))

(112) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31))
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(113) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P61_IN_G(s(T31)) → U16_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))

(114) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P82_IN_G(T31) → U18_G(T31, less86_in_g(T31))
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(115) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P82_IN_G(T31) → U18_G(T31, less86_in_g(T31)) at position [1] we obtained the following new rules [LPAR04]:

P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))

(116) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less86_in_g(s(T39)) → less86_out_g(s(T39))
less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(117) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(118) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less86_in_g(x0)
less152_in_ag(x0)
U20_ag(x0, x1)

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

(119) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

less86_in_g(x0)

(120) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(121) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(T42))) → P16_IN_G(T42) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

(122) Obligation:

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

U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(123) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(T56))))))) → U8_AG(T56, less152_in_ag(T56)) at position [1] we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))

(124) Obligation:

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

P1_IN_AG(s(s(T42))) → P16_IN_G(T42)
U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(125) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(T42))) → P16_IN_G(T42) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

(126) Obligation:

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

U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(127) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U10_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(128) Obligation:

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

P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45)
U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(129) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(T45)))) → P28_IN_G(T45) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))

(130) Obligation:

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

U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(131) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(132) Obligation:

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

P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48)
U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(133) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(T48))))) → P43_IN_G(T48) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))

(134) Obligation:

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

U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(135) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(136) Obligation:

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

P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51)
U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(137) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(T51)))))) → P61_IN_G(T51) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))

(138) Obligation:

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

U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(139) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(140) Obligation:

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

P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31)))))))
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(141) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U18_G(T31, less86_out_g(T31)) → P1_IN_AG(s(s(s(s(s(s(T31))))))) we obtained the following new rules [LPAR04]:

U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(142) Obligation:

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

P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56)))))))
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(143) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U8_AG(T56, less152_out_ag(T55, T56)) → P1_IN_AG(s(s(s(s(s(s(T56))))))) we obtained the following new rules [LPAR04]:

U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

(144) Obligation:

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

P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54)
P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(145) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(T54))))))) → P82_IN_G(T54) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))

(146) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0)))))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(147) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P16_IN_G(s(s(s(s(z0))))) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

(148) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(149) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P28_IN_G(s(s(s(z0)))) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))

(150) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0)))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(151) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P43_IN_G(s(s(z0))) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))

(152) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(153) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_AG(s(s(s(s(s(s(z0))))))) → P61_IN_G(s(z0)) we obtained the following new rules [LPAR04]:

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))

(154) Obligation:

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

P16_IN_G(s(s(s(s(s(x0)))))) → U10_G(s(x0), less86_out_g(s(x0)))
P28_IN_G(s(s(s(s(x0))))) → U12_G(s(x0), less86_out_g(s(x0)))
P43_IN_G(s(s(s(x0)))) → U14_G(s(x0), less86_out_g(s(x0)))
P61_IN_G(s(s(x0))) → U16_G(s(x0), less86_out_g(s(x0)))
P82_IN_G(s(x0)) → U18_G(s(x0), less86_out_g(s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))
P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), U20_ag(x0, less152_in_ag(x0)))
U10_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P28_IN_G(s(s(s(s(z0)))))
U12_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P43_IN_G(s(s(s(z0))))
U14_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P61_IN_G(s(s(z0)))
U16_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U18_G(s(z0), less86_out_g(s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
U8_AG(s(z0), less152_out_ag(x1, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P82_IN_G(s(z0))
P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → P16_IN_G(s(s(s(s(s(z0))))))

The TRS R consists of the following rules:

less152_in_ag(s(T58)) → less152_out_ag(0, s(T58))
less152_in_ag(s(T60)) → U20_ag(T60, less152_in_ag(T60))
U20_ag(T60, less152_out_ag(T59, T60)) → less152_out_ag(s(T59), s(T60))

The set Q consists of the following terms:

less152_in_ag(x0)
U20_ag(x0, x1)

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

(155) NonTerminationProof (EQUIVALENT transformation)

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

s = U8_AG(s(z0), less152_out_ag(0, s(z0))) evaluates to t =U8_AG(s(z0), less152_out_ag(0, s(z0)))

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

• Matcher: [ ]
• Semiunifier: [ ]

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Rewriting sequence

U8_AG(s(z0), less152_out_ag(0, s(z0))) → P1_IN_AG(s(s(s(s(s(s(s(z0))))))))
with rule U8_AG(s(z0'), less152_out_ag(0, s(z0'))) → P1_IN_AG(s(s(s(s(s(s(s(z0')))))))) at position [] and matcher [z0' / z0]

P1_IN_AG(s(s(s(s(s(s(s(z0)))))))) → U8_AG(s(z0), less152_out_ag(0, s(z0)))
with rule P1_IN_AG(s(s(s(s(s(s(s(x0)))))))) → U8_AG(s(x0), less152_out_ag(0, s(x0)))

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


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

(156) FALSE