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

(0) Obligation:

Clauses:

div(0, Y, 0) :- no(zero(Y)).
div(X, Y, s(Z)) :- ','(no(zero(X)), ','(no(zero(Y)), ','(minus(X, Y, U), div(U, Y, Z)))).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: div(g,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: no(zero(T5)) | div(0, T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: div(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX3 is free\ndiv(T1, T2, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: no(zero(T5)), SCOPE: 2, CLAUSE: 6 | no(zero(T5)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(call(zero(T8)), ','(!_2, failure(a))) | no(zero(T8)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(zero(T8), ','(!_2, failure(a))) | ?_3 | no(zero(T8)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(zero(T8), ','(!_2, failure(a))), SCOPE: 4, CLAUSE: 5 | ?_4 | ?_3 | no(zero(T8)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(!_2, failure(a)) | ?_4 | ?_3 | no(zero(0)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ?_4 | ?_3 | no(zero(T8)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: failure(a) | div(0, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: div(0, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: failure(a), SCOPE: 5, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12))))\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 6, CLAUSE: 6 | ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 6, CLAUSE: 7\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(call(zero(0)), ','(!_6, failure(a))), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))) | ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 6, CLAUSE: 7\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(zero(0), ','(','(!_6, failure(a)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12))))) | ?_7 | ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 6, CLAUSE: 7\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(zero(0), ','(','(!_6, failure(a)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12))))), SCOPE: 8, CLAUSE: 5 | ?_8 | ?_7 | ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 6, CLAUSE: 7\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(!_6, failure(a)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))) | ?_8 | ?_7 | ','(no(zero(0)), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 6, CLAUSE: 7\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(failure(a), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12))))\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(failure(a), ','(no(zero(0)), ','(minus(0, 0, X17), div(X17, 0, T12)))), SCOPE: 9, CLAUSE: 8\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ?_3 | no(zero(T8)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: no(zero(T8)), SCOPE: 2, CLAUSE: 7 | ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: no(zero(T8)), SCOPE: 2, CLAUSE: 7\n\nT8 is ground\nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ?_2 | div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: div(0, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24))))\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 10, CLAUSE: 6 | ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 10, CLAUSE: 7\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(','(call(zero(0)), ','(!_10, failure(a))), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))) | ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 10, CLAUSE: 7\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(zero(0), ','(','(!_10, failure(a)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24))))) | ?_11 | ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 10, CLAUSE: 7\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(zero(0), ','(','(!_10, failure(a)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24))))), SCOPE: 12, CLAUSE: 5 | ?_12 | ?_11 | ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 10, CLAUSE: 7\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(','(!_10, failure(a)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))) | ?_12 | ?_11 | ','(no(zero(0)), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 10, CLAUSE: 7\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(failure(a), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24))))\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(failure(a), ','(no(zero(T22)), ','(minus(0, T22, X40), div(X40, T22, T24)))), SCOPE: 13, CLAUSE: 8\n\nT22 is ground\nX40 is free\nzero(T22) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(no(zero(T32)), ','(no(zero(T33)), ','(minus(T32, T33, X54), div(X54, T33, T35))))\n\nT32 is ground\nT33 is ground\nX3 is free\nX54 is free\ndiv(T32, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(no(zero(T32)), ','(no(zero(T33)), ','(minus(T32, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 6 | ','(no(zero(T32)), ','(no(zero(T33)), ','(minus(T32, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT32 is ground\nT33 is ground\nX3 is free\nX54 is free\ndiv(T32, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(','(call(zero(T38)), ','(!_14, failure(a))), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))) | ','(no(zero(T38)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nT38 is ground\nX3 is free\nX54 is free\ndiv(T38, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(zero(T38), ','(','(!_14, failure(a)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35))))) | ?_15 | ','(no(zero(T38)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nT38 is ground\nX3 is free\nX54 is free\ndiv(T38, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(zero(T38), ','(','(!_14, failure(a)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35))))), SCOPE: 16, CLAUSE: 5 | ?_16 | ?_15 | ','(no(zero(T38)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nT38 is ground\nX3 is free\nX54 is free\ndiv(T38, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(','(!_14, failure(a)), ','(no(zero(T33)), ','(minus(0, T33, X54), div(X54, T33, T35)))) | ?_16 | ?_15 | ','(no(zero(0)), ','(no(zero(T33)), ','(minus(0, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nX3 is free\nX54 is free\ndiv(0, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ?_16 | ?_15 | ','(no(zero(T38)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nT38 is ground\nX3 is free\nX54 is free\ndiv(T38, T33, T3) !=? div(0, X3, 0)\nzero(T38) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(failure(a), ','(no(zero(T33)), ','(minus(0, T33, X54), div(X54, T33, T35))))\n\nT33 is ground\nX3 is free\nX54 is free\ndiv(0, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(no(zero(T33)), ','(minus(0, T33, X54), div(X54, T33, T35)))), SCOPE: 17, CLAUSE: 8\n\nT33 is ground\nX3 is free\nX54 is free\ndiv(0, T33, T3) !=? div(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ?_15 | ','(no(zero(T38)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nT38 is ground\nX3 is free\nX54 is free\ndiv(T38, T33, T3) !=? div(0, X3, 0)\nzero(T38) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(no(zero(T38)), ','(no(zero(T33)), ','(minus(T38, T33, X54), div(X54, T33, T35)))), SCOPE: 14, CLAUSE: 7\n\nT33 is ground\nT38 is ground\nX3 is free\nX54 is free\ndiv(T38, T33, T3) !=? div(0, X3, 0)\nzero(T38) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(zero(T33)), ','(minus(T47, T33, X54), div(X54, T33, T35)))\n\nT33 is ground\nT47 is ground\nX3 is free\nX54 is free\ndiv(T47, T33, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(no(zero(T33)), ','(minus(T47, T33, X54), div(X54, T33, T35))), SCOPE: 18, CLAUSE: 6 | ','(no(zero(T33)), ','(minus(T47, T33, X54), div(X54, T33, T35))), SCOPE: 18, CLAUSE: 7\n\nT33 is ground\nT47 is ground\nX3 is free\nX54 is free\ndiv(T47, T33, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(','(call(zero(T54)), ','(!_18, failure(a))), ','(minus(T47, T54, X54), div(X54, T54, T35))) | ','(no(zero(T54)), ','(minus(T47, T54, X54), div(X54, T54, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nT54 is ground\nX3 is free\nX54 is free\ndiv(T47, T54, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(zero(T54), ','(','(!_18, failure(a)), ','(minus(T47, T54, X54), div(X54, T54, T35)))) | ?_19 | ','(no(zero(T54)), ','(minus(T47, T54, X54), div(X54, T54, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nT54 is ground\nX3 is free\nX54 is free\ndiv(T47, T54, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(zero(T54), ','(','(!_18, failure(a)), ','(minus(T47, T54, X54), div(X54, T54, T35)))), SCOPE: 20, CLAUSE: 5 | ?_20 | ?_19 | ','(no(zero(T54)), ','(minus(T47, T54, X54), div(X54, T54, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nT54 is ground\nX3 is free\nX54 is free\ndiv(T47, T54, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(','(!_18, failure(a)), ','(minus(T47, 0, X54), div(X54, 0, T35))) | ?_20 | ?_19 | ','(no(zero(0)), ','(minus(T47, 0, X54), div(X54, 0, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nX3 is free\nX54 is free\ndiv(T47, 0, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ?_20 | ?_19 | ','(no(zero(T54)), ','(minus(T47, T54, X54), div(X54, T54, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nT54 is ground\nX3 is free\nX54 is free\ndiv(T47, T54, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T54) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(failure(a), ','(minus(T47, 0, X54), div(X54, 0, T35)))\n\nT47 is ground\nX3 is free\nX54 is free\ndiv(T47, 0, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(failure(a), ','(minus(T47, 0, X54), div(X54, 0, T35))), SCOPE: 21, CLAUSE: 8\n\nT47 is ground\nX3 is free\nX54 is free\ndiv(T47, 0, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ?_19 | ','(no(zero(T54)), ','(minus(T47, T54, X54), div(X54, T54, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nT54 is ground\nX3 is free\nX54 is free\ndiv(T47, T54, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T54) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(no(zero(T54)), ','(minus(T47, T54, X54), div(X54, T54, T35))), SCOPE: 18, CLAUSE: 7\n\nT47 is ground\nT54 is ground\nX3 is free\nX54 is free\ndiv(T47, T54, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T54) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(minus(T47, T64, X54), div(X54, T64, T35))\n\nT47 is ground\nT64 is ground\nX3 is free\nX54 is free\ndiv(T47, T64, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T64) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: minus(T47, T64, X54)\n\nT47 is ground\nT64 is ground\nX3 is free\nX54 is free\ndiv(T47, T64, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T64) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: div(T71, T64, T35)\n\nT47 is ground\nT64 is ground\nT71 is ground\nX3 is free\ndiv(T47, T64, T3) !=? div(0, X3, 0)\nzero(T64) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: minus(T47, T64, X54), SCOPE: 22, CLAUSE: 2 | minus(T47, T64, X54), SCOPE: 22, CLAUSE: 3 | minus(T47, T64, X54), SCOPE: 22, CLAUSE: 4\n\nT47 is ground\nT64 is ground\nX3 is free\nX54 is free\ndiv(T47, T64, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T64) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: minus(T47, T64, X54), SCOPE: 22, CLAUSE: 3 | minus(T47, T64, X54), SCOPE: 22, CLAUSE: 4\n\nT47 is ground\nT64 is ground\nX3 is free\nX54 is free\ndiv(T47, T64, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T64) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: minus(T47, T64, X54), SCOPE: 22, CLAUSE: 4\n\nT47 is ground\nT64 is ground\nX3 is free\nX54 is free\ndiv(T47, T64, T3) !=? div(0, X3, 0)\nzero(T47) !=? zero(0)\nzero(T64) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: minus(T84, T85, X118)\n\nT84 is ground\nT85 is ground\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: minus(T84, T85, X118), SCOPE: 23, CLAUSE: 2 | minus(T84, T85, X118), SCOPE: 23, CLAUSE: 3 | minus(T84, T85, X118), SCOPE: 23, CLAUSE: 4\n\nT84 is ground\nT85 is ground\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: minus(T84, T85, X118), SCOPE: 23, CLAUSE: 2\n\nT84 is ground\nT85 is ground\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: minus(T84, T85, X118), SCOPE: 23, CLAUSE: 3 | minus(T84, T85, X118), SCOPE: 23, CLAUSE: 4\n\nT84 is ground\nT85 is ground\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: minus(T84, T85, X118), SCOPE: 23, CLAUSE: 3\n\nT84 is ground\nT85 is ground\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: minus(T84, T85, X118), SCOPE: 23, CLAUSE: 4\n\nT84 is ground\nT85 is ground\nX118 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: minus(T102, T103, X142)\n\nT102 is ground\nT103 is ground\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 89 [arrowhead = none ];
89 -> 2;

90 [label="EVAL with clause\ndiv(0, X3, 0) :- no(zero(X3)).\nand substitutionT1 -> 0,\nT2 -> T5,\nX3 -> T5,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 90 [arrowhead = none ];
90 -> 3;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 91 [arrowhead = none ];
91 -> 4;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 92 [arrowhead = none ];
92 -> 5;

93 [label="EVAL with clause\ndiv(X51, X52, s(X53)) :- ','(no(zero(X51)), ','(no(zero(X52)), ','(minus(X51, X52, X54), div(X54, X52, X53)))).\nand substitutionT1 -> T32,\nX51 -> T32,\nT2 -> T33,\nX52 -> T33,\nX53 -> T35,\nT3 -> s(T35),\nT34 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 93 [arrowhead = none ];
93 -> 43;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 94 [arrowhead = none ];
94 -> 44;

95 [label="ONLY EVAL with clause\nno(X6) :- ','(call(X6), ','(!_2, failure(a))).\nand substitutionT5 -> T8,\nX6 -> zero(T8)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 95 [arrowhead = none ];
95 -> 6;

96 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 96 [arrowhead = none ];
96 -> 7;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 97 [arrowhead = none ];
97 -> 8;

98 [label="EVAL with clause\nzero(0).\nand substitutionT8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 98 [arrowhead = none ];
98 -> 9;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 99 [arrowhead = none ];
99 -> 10;

100 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 100 [arrowhead = none ];
100 -> 11;

101 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 101 [arrowhead = none ];
101 -> 26;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 102 [arrowhead = none ];
102 -> 12;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 103 [arrowhead = none ];
103 -> 13;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 104 [arrowhead = none ];
104 -> 14;

105 [label="EVAL with clause\ndiv(X14, X15, s(X16)) :- ','(no(zero(X14)), ','(no(zero(X15)), ','(minus(X14, X15, X17), div(X17, X15, X16)))).\nand substitutionX14 -> 0,\nX15 -> 0,\nX16 -> T12,\nT3 -> s(T12),\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 105 [arrowhead = none ];
105 -> 16;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 106 [arrowhead = none ];
106 -> 17;

107 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 107 [arrowhead = none ];
107 -> 15;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 108 [arrowhead = none ];
108 -> 18;

109 [label="ONLY EVAL with clause\nno(X22) :- ','(call(X22), ','(!_6, failure(a))).\nand substitutionX22 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 109 [arrowhead = none ];
109 -> 19;

110 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 110 [arrowhead = none ];
110 -> 20;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 111 [arrowhead = none ];
111 -> 21;

112 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 112 [arrowhead = none ];
112 -> 22;

113 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 113 [arrowhead = none ];
113 -> 23;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 114 [arrowhead = none ];
114 -> 24;

115 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 115 [arrowhead = none ];
115 -> 25;

116 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 116 [arrowhead = none ];
116 -> 27;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 117 [arrowhead = none ];
117 -> 28;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 118 [arrowhead = none ];
118 -> 29;

119 [label="ONLY EVAL with clause\nno(X29).\nand substitutionT8 -> T17,\nX29 -> zero(T17)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 119 [arrowhead = none ];
119 -> 30;

120 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 120 [arrowhead = none ];
120 -> 32;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 121 [arrowhead = none ];
121 -> 31;

122 [label="EVAL with clause\ndiv(X37, X38, s(X39)) :- ','(no(zero(X37)), ','(no(zero(X38)), ','(minus(X37, X38, X40), div(X40, X38, X39)))).\nand substitutionX37 -> 0,\nT8 -> T22,\nX38 -> T22,\nX39 -> T24,\nT3 -> s(T24),\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 122 [arrowhead = none ];
122 -> 33;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 123 [arrowhead = none ];
123 -> 34;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 124 [arrowhead = none ];
124 -> 35;

125 [label="ONLY EVAL with clause\nno(X45) :- ','(call(X45), ','(!_10, failure(a))).\nand substitutionX45 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 125 [arrowhead = none ];
125 -> 36;

126 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 126 [arrowhead = none ];
126 -> 37;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 127 [arrowhead = none ];
127 -> 38;

128 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 128 [arrowhead = none ];
128 -> 39;

129 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 129 [arrowhead = none ];
129 -> 40;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 130 [arrowhead = none ];
130 -> 41;

131 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 131 [arrowhead = none ];
131 -> 42;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 132 [arrowhead = none ];
132 -> 45;

133 [label="ONLY EVAL with clause\nno(X57) :- ','(call(X57), ','(!_14, failure(a))).\nand substitutionT32 -> T38,\nX57 -> zero(T38)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 133 [arrowhead = none ];
133 -> 46;

134 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 134 [arrowhead = none ];
134 -> 47;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 135 [arrowhead = none ];
135 -> 48;

136 [label="EVAL with clause\nzero(0).\nand substitutionT38 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 136 [arrowhead = none ];
136 -> 49;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 137 [arrowhead = none ];
137 -> 50;

138 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 139 [arrowhead = none ];
139 -> 54;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 140 [arrowhead = none ];
140 -> 52;

141 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 141 [arrowhead = none ];
141 -> 53;

142 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 142 [arrowhead = none ];
142 -> 55;

143 [label="ONLY EVAL with clause\nno(X66).\nand substitutionT38 -> T47,\nX66 -> zero(T47)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 143 [arrowhead = none ];
143 -> 56;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 144 [arrowhead = none ];
144 -> 57;

145 [label="ONLY EVAL with clause\nno(X73) :- ','(call(X73), ','(!_18, failure(a))).\nand substitutionT33 -> T54,\nX73 -> zero(T54)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 145 [arrowhead = none ];
145 -> 58;

146 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 146 [arrowhead = none ];
146 -> 59;

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 147 [arrowhead = none ];
147 -> 60;

148 [label="EVAL with clause\nzero(0).\nand substitutionT54 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 148 [arrowhead = none ];
148 -> 61;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 149 [arrowhead = none ];
149 -> 62;

150 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 150 [arrowhead = none ];
150 -> 63;

151 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 151 [arrowhead = none ];
151 -> 66;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 152 [arrowhead = none ];
152 -> 64;

153 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 153 [arrowhead = none ];
153 -> 65;

154 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 154 [arrowhead = none ];
154 -> 67;

155 [label="ONLY EVAL with clause\nno(X88).\nand substitutionT54 -> T64,\nX88 -> zero(T64)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 155 [arrowhead = none ];
155 -> 68;

156 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
68 -> 156 [arrowhead = none ];
156 -> 69;

157 [label="SPLIT 2\nnew knowledge:\nT47 is ground\nT64 is ground\nT71 is ground\nreplacements:X54 -> T71", fontsize=14, style = filled, fillcolor = violet];
68 -> 157 [arrowhead = none ];
157 -> 70;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 158 [arrowhead = none ];
158 -> 71;

159 [label="INSTANCE with matching:\nT1 -> T71\nT2 -> T64\nT3 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 159 [arrowhead = none , style=dashed];
159 -> 1[style=dashed];

160 [label="BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (zero(T47), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 160 [arrowhead = none ];
160 -> 72;

161 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (zero(T64), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 161 [arrowhead = none ];
161 -> 73;

162 [label="EVAL with clause\nminus(s(X115), s(X116), X117) :- minus(X115, X116, X117).\nand substitutionX115 -> T84,\nT47 -> s(T84),\nX116 -> T85,\nT64 -> s(T85),\nX54 -> X118,\nX117 -> X118", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 162 [arrowhead = none ];
162 -> 74;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 163 [arrowhead = none ];
163 -> 75;

164 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 164 [arrowhead = none ];
164 -> 76;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 165 [arrowhead = none ];
165 -> 77;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 166 [arrowhead = none ];
166 -> 78;

167 [label="EVAL with clause\nminus(0, X125, 0).\nand substitutionT84 -> 0,\nT85 -> T92,\nX125 -> T92,\nX118 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 167 [arrowhead = none ];
167 -> 79;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 168 [arrowhead = none ];
168 -> 80;

169 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
78 -> 169 [arrowhead = none ];
169 -> 82;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
78 -> 170 [arrowhead = none ];
170 -> 83;

171 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 171 [arrowhead = none ];
171 -> 81;

172 [label="EVAL with clause\nminus(X130, 0, X130).\nand substitutionT84 -> T97,\nX130 -> T97,\nT85 -> 0,\nX118 -> T97", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 172 [arrowhead = none ];
172 -> 84;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 173 [arrowhead = none ];
173 -> 85;

174 [label="EVAL with clause\nminus(s(X139), s(X140), X141) :- minus(X139, X140, X141).\nand substitutionX139 -> T102,\nT84 -> s(T102),\nX140 -> T103,\nT85 -> s(T103),\nX118 -> X142,\nX141 -> X142", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 174 [arrowhead = none ];
174 -> 87;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 175 [arrowhead = none ];
175 -> 88;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 176 [arrowhead = none ];
176 -> 86;

177 [label="INSTANCE with matching:\nT84 -> T102\nT85 -> T103\nX118 -> X142", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 177 [arrowhead = none , style=dashed];
177 -> 74[style=dashed];

}

(2) Obligation:

Clauses:

minusA(0, T92, 0).
minusA(T97, 0, T97).
minusA(s(T102), s(T103), X142) :- minusA(T102, T103, X142).
minusB(s(T84), s(T85), X118) :- minusA(T84, T85, X118).
divC(0, T17, 0).
divC(T47, T64, s(T35)) :- minusB(T47, T64, X54).
divC(T47, T64, s(T35)) :- ','(minusB(T47, T64, T71), divC(T71, T64, T35)).

Query: divC(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
divC_in: (b,b,f)
minusB_in: (b,b,f)
minusA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

divC_in_gga(0, T17, 0) → divC_out_gga(0, T17, 0)
divC_in_gga(T47, T64, s(T35)) → U3_gga(T47, T64, T35, minusB_in_gga(T47, T64, X54))
minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
U3_gga(T47, T64, T35, minusB_out_gga(T47, T64, X54)) → divC_out_gga(T47, T64, s(T35))
divC_in_gga(T47, T64, s(T35)) → U4_gga(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_gga(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_gga(T47, T64, T35, divC_in_gga(T71, T64, T35))
U5_gga(T47, T64, T35, divC_out_gga(T71, T64, T35)) → divC_out_gga(T47, T64, s(T35))

The argument filtering Pi contains the following mapping:
divC_in_gga(x1, x2, x3)  =  divC_in_gga(x1, x2)
0  =  0
divC_out_gga(x1, x2, x3)  =  divC_out_gga
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

divC_in_gga(0, T17, 0) → divC_out_gga(0, T17, 0)
divC_in_gga(T47, T64, s(T35)) → U3_gga(T47, T64, T35, minusB_in_gga(T47, T64, X54))
minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
U3_gga(T47, T64, T35, minusB_out_gga(T47, T64, X54)) → divC_out_gga(T47, T64, s(T35))
divC_in_gga(T47, T64, s(T35)) → U4_gga(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_gga(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_gga(T47, T64, T35, divC_in_gga(T71, T64, T35))
U5_gga(T47, T64, T35, divC_out_gga(T71, T64, T35)) → divC_out_gga(T47, T64, s(T35))

The argument filtering Pi contains the following mapping:
divC_in_gga(x1, x2, x3)  =  divC_in_gga(x1, x2)
0  =  0
divC_out_gga(x1, x2, x3)  =  divC_out_gga
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

DIVC_IN_GGA(T47, T64, s(T35)) → U3_GGA(T47, T64, T35, minusB_in_gga(T47, T64, X54))
DIVC_IN_GGA(T47, T64, s(T35)) → MINUSB_IN_GGA(T47, T64, X54)
MINUSB_IN_GGA(s(T84), s(T85), X118) → U2_GGA(T84, T85, X118, minusA_in_gga(T84, T85, X118))
MINUSB_IN_GGA(s(T84), s(T85), X118) → MINUSA_IN_GGA(T84, T85, X118)
MINUSA_IN_GGA(s(T102), s(T103), X142) → U1_GGA(T102, T103, X142, minusA_in_gga(T102, T103, X142))
MINUSA_IN_GGA(s(T102), s(T103), X142) → MINUSA_IN_GGA(T102, T103, X142)
DIVC_IN_GGA(T47, T64, s(T35)) → U4_GGA(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_GGA(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_GGA(T47, T64, T35, divC_in_gga(T71, T64, T35))
U4_GGA(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → DIVC_IN_GGA(T71, T64, T35)

The TRS R consists of the following rules:

divC_in_gga(0, T17, 0) → divC_out_gga(0, T17, 0)
divC_in_gga(T47, T64, s(T35)) → U3_gga(T47, T64, T35, minusB_in_gga(T47, T64, X54))
minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
U3_gga(T47, T64, T35, minusB_out_gga(T47, T64, X54)) → divC_out_gga(T47, T64, s(T35))
divC_in_gga(T47, T64, s(T35)) → U4_gga(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_gga(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_gga(T47, T64, T35, divC_in_gga(T71, T64, T35))
U5_gga(T47, T64, T35, divC_out_gga(T71, T64, T35)) → divC_out_gga(T47, T64, s(T35))

The argument filtering Pi contains the following mapping:
divC_in_gga(x1, x2, x3)  =  divC_in_gga(x1, x2)
0  =  0
divC_out_gga(x1, x2, x3)  =  divC_out_gga
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
DIVC_IN_GGA(x1, x2, x3)  =  DIVC_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
MINUSB_IN_GGA(x1, x2, x3)  =  MINUSB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
MINUSA_IN_GGA(x1, x2, x3)  =  MINUSA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(6) Obligation:

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

DIVC_IN_GGA(T47, T64, s(T35)) → U3_GGA(T47, T64, T35, minusB_in_gga(T47, T64, X54))
DIVC_IN_GGA(T47, T64, s(T35)) → MINUSB_IN_GGA(T47, T64, X54)
MINUSB_IN_GGA(s(T84), s(T85), X118) → U2_GGA(T84, T85, X118, minusA_in_gga(T84, T85, X118))
MINUSB_IN_GGA(s(T84), s(T85), X118) → MINUSA_IN_GGA(T84, T85, X118)
MINUSA_IN_GGA(s(T102), s(T103), X142) → U1_GGA(T102, T103, X142, minusA_in_gga(T102, T103, X142))
MINUSA_IN_GGA(s(T102), s(T103), X142) → MINUSA_IN_GGA(T102, T103, X142)
DIVC_IN_GGA(T47, T64, s(T35)) → U4_GGA(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_GGA(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_GGA(T47, T64, T35, divC_in_gga(T71, T64, T35))
U4_GGA(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → DIVC_IN_GGA(T71, T64, T35)

The TRS R consists of the following rules:

divC_in_gga(0, T17, 0) → divC_out_gga(0, T17, 0)
divC_in_gga(T47, T64, s(T35)) → U3_gga(T47, T64, T35, minusB_in_gga(T47, T64, X54))
minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
U3_gga(T47, T64, T35, minusB_out_gga(T47, T64, X54)) → divC_out_gga(T47, T64, s(T35))
divC_in_gga(T47, T64, s(T35)) → U4_gga(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_gga(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_gga(T47, T64, T35, divC_in_gga(T71, T64, T35))
U5_gga(T47, T64, T35, divC_out_gga(T71, T64, T35)) → divC_out_gga(T47, T64, s(T35))

The argument filtering Pi contains the following mapping:
divC_in_gga(x1, x2, x3)  =  divC_in_gga(x1, x2)
0  =  0
divC_out_gga(x1, x2, x3)  =  divC_out_gga
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
DIVC_IN_GGA(x1, x2, x3)  =  DIVC_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
MINUSB_IN_GGA(x1, x2, x3)  =  MINUSB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
MINUSA_IN_GGA(x1, x2, x3)  =  MINUSA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MINUSA_IN_GGA(s(T102), s(T103), X142) → MINUSA_IN_GGA(T102, T103, X142)

The TRS R consists of the following rules:

divC_in_gga(0, T17, 0) → divC_out_gga(0, T17, 0)
divC_in_gga(T47, T64, s(T35)) → U3_gga(T47, T64, T35, minusB_in_gga(T47, T64, X54))
minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
U3_gga(T47, T64, T35, minusB_out_gga(T47, T64, X54)) → divC_out_gga(T47, T64, s(T35))
divC_in_gga(T47, T64, s(T35)) → U4_gga(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_gga(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_gga(T47, T64, T35, divC_in_gga(T71, T64, T35))
U5_gga(T47, T64, T35, divC_out_gga(T71, T64, T35)) → divC_out_gga(T47, T64, s(T35))

The argument filtering Pi contains the following mapping:
divC_in_gga(x1, x2, x3)  =  divC_in_gga(x1, x2)
0  =  0
divC_out_gga(x1, x2, x3)  =  divC_out_gga
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
MINUSA_IN_GGA(x1, x2, x3)  =  MINUSA_IN_GGA(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:

MINUSA_IN_GGA(s(T102), s(T103), X142) → MINUSA_IN_GGA(T102, T103, X142)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

MINUSA_IN_GGA(s(T102), s(T103)) → MINUSA_IN_GGA(T102, T103)

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:

• MINUSA_IN_GGA(s(T102), s(T103)) → MINUSA_IN_GGA(T102, T103)
  The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

DIVC_IN_GGA(T47, T64, s(T35)) → U4_GGA(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_GGA(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → DIVC_IN_GGA(T71, T64, T35)

The TRS R consists of the following rules:

divC_in_gga(0, T17, 0) → divC_out_gga(0, T17, 0)
divC_in_gga(T47, T64, s(T35)) → U3_gga(T47, T64, T35, minusB_in_gga(T47, T64, X54))
minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
U3_gga(T47, T64, T35, minusB_out_gga(T47, T64, X54)) → divC_out_gga(T47, T64, s(T35))
divC_in_gga(T47, T64, s(T35)) → U4_gga(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_gga(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → U5_gga(T47, T64, T35, divC_in_gga(T71, T64, T35))
U5_gga(T47, T64, T35, divC_out_gga(T71, T64, T35)) → divC_out_gga(T47, T64, s(T35))

The argument filtering Pi contains the following mapping:
divC_in_gga(x1, x2, x3)  =  divC_in_gga(x1, x2)
0  =  0
divC_out_gga(x1, x2, x3)  =  divC_out_gga
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
DIVC_IN_GGA(x1, x2, x3)  =  DIVC_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)

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:

DIVC_IN_GGA(T47, T64, s(T35)) → U4_GGA(T47, T64, T35, minusB_in_gga(T47, T64, T71))
U4_GGA(T47, T64, T35, minusB_out_gga(T47, T64, T71)) → DIVC_IN_GGA(T71, T64, T35)

The TRS R consists of the following rules:

minusB_in_gga(s(T84), s(T85), X118) → U2_gga(T84, T85, X118, minusA_in_gga(T84, T85, X118))
U2_gga(T84, T85, X118, minusA_out_gga(T84, T85, X118)) → minusB_out_gga(s(T84), s(T85), X118)
minusA_in_gga(0, T92, 0) → minusA_out_gga(0, T92, 0)
minusA_in_gga(T97, 0, T97) → minusA_out_gga(T97, 0, T97)
minusA_in_gga(s(T102), s(T103), X142) → U1_gga(T102, T103, X142, minusA_in_gga(T102, T103, X142))
U1_gga(T102, T103, X142, minusA_out_gga(T102, T103, X142)) → minusA_out_gga(s(T102), s(T103), X142)

The argument filtering Pi contains the following mapping:
0  =  0
minusB_in_gga(x1, x2, x3)  =  minusB_in_gga(x1, x2)
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3)  =  minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minusB_out_gga(x1, x2, x3)  =  minusB_out_gga(x3)
DIVC_IN_GGA(x1, x2, x3)  =  DIVC_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)

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:

DIVC_IN_GGA(T47, T64) → U4_GGA(T64, minusB_in_gga(T47, T64))
U4_GGA(T64, minusB_out_gga(T71)) → DIVC_IN_GGA(T71, T64)

The TRS R consists of the following rules:

minusB_in_gga(s(T84), s(T85)) → U2_gga(minusA_in_gga(T84, T85))
U2_gga(minusA_out_gga(X118)) → minusB_out_gga(X118)
minusA_in_gga(0, T92) → minusA_out_gga(0)
minusA_in_gga(T97, 0) → minusA_out_gga(T97)
minusA_in_gga(s(T102), s(T103)) → U1_gga(minusA_in_gga(T102, T103))
U1_gga(minusA_out_gga(X142)) → minusA_out_gga(X142)

The set Q consists of the following terms:

minusB_in_gga(x0, x1)
U2_gga(x0)
minusA_in_gga(x0, x1)
U1_gga(x0)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

U4_GGA(T64, minusB_out_gga(T71)) → DIVC_IN_GGA(T71, T64)

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

POL(0) = 0   
POL(DIVC_IN_GGA(x₁, x₂)) = 1 + x₁   
POL(U1_gga(x₁)) = x₁   
POL(U2_gga(x₁)) = x₁   
POL(U4_GGA(x₁, x₂)) = 1 + x₂   
POL(minusA_in_gga(x₁, x₂)) = 1 + x₁   
POL(minusA_out_gga(x₁)) = 1 + x₁   
POL(minusB_in_gga(x₁, x₂)) = x₁   
POL(minusB_out_gga(x₁)) = 1 + x₁   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

minusB_in_gga(s(T84), s(T85)) → U2_gga(minusA_in_gga(T84, T85))
minusA_in_gga(0, T92) → minusA_out_gga(0)
minusA_in_gga(T97, 0) → minusA_out_gga(T97)
minusA_in_gga(s(T102), s(T103)) → U1_gga(minusA_in_gga(T102, T103))
U2_gga(minusA_out_gga(X118)) → minusB_out_gga(X118)
U1_gga(minusA_out_gga(X142)) → minusA_out_gga(X142)

(22) Obligation:

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

DIVC_IN_GGA(T47, T64) → U4_GGA(T64, minusB_in_gga(T47, T64))

The TRS R consists of the following rules:

minusB_in_gga(s(T84), s(T85)) → U2_gga(minusA_in_gga(T84, T85))
U2_gga(minusA_out_gga(X118)) → minusB_out_gga(X118)
minusA_in_gga(0, T92) → minusA_out_gga(0)
minusA_in_gga(T97, 0) → minusA_out_gga(T97)
minusA_in_gga(s(T102), s(T103)) → U1_gga(minusA_in_gga(T102, T103))
U1_gga(minusA_out_gga(X142)) → minusA_out_gga(X142)

The set Q consists of the following terms:

minusB_in_gga(x0, x1)
U2_gga(x0)
minusA_in_gga(x0, x1)
U1_gga(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE