Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

minus(X, Y, Z) :- ','(=(X, 0), ','(!, =(Z, 0))).
minus(X, Y, Z) :- ','(=(Y, 0), ','(!, =(Z, X))).
minus(X, Y, Z) :- ','(=(X, s(A)), ','(=(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=(Y, 0), ','(!, fail)).
div(X, Y, Z) :- ','(=(X, 0), ','(!, =(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =(Z, s(V)))).
=(X, X).

Queries:

div(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

Clauses:

minus(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
minus(X, Y, Z) :- ','(=1(Y, 0), ','(!, =1(Z, X))).
minus(X, Y, Z) :- ','(=1(X, s(A)), ','(=1(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=1(Y, 0), ','(!, fail)).
div(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =1(Z, s(V)))).
=1(X, X).
fail :- fail(b).
fail(a).

Queries:

div(g,g,a).

(3) BuiltinConflictTransformerProof (SOUND transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(4) Obligation:

Clauses:

minus(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
minus(X, Y, Z) :- ','(=1(Y, 0), ','(!, =1(Z, X))).
minus(X, Y, Z) :- ','(=1(X, s(A)), ','(=1(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=1(Y, 0), ','(!, user_defined_fail)).
div(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =1(Z, s(V)))).
=1(X, X).
user_defined_fail :- fail(b).
fail(a).

Queries:

div(g,g,a).

(5) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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: 3 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(=1(T8, 0), ','(!_1, user_defined_fail)) | div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(=1(T8, 0), ','(!_1, user_defined_fail)), SCOPE: 2, CLAUSE: 6 | ?_2 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, user_defined_fail) | ?_2 | div(T7, 0, T3), SCOPE: 1, CLAUSE: 4 | div(T7, 0, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ?_2 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground\n=1(T8, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: user_defined_fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: user_defined_fail, SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: fail(b)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: fail(b), SCOPE: 4, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground\n=1(T8, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(=1(T20, 0), ','(!_1, =1(T23, 0))) | div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\n=1(T21, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(=1(T20, 0), ','(!_1, =1(T23, 0))), SCOPE: 5, CLAUSE: 6 | ?_5 | div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\n=1(T21, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(!_1, =1(T23, 0)) | ?_5 | div(0, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT21 is ground\n=1(T21, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ?_5 | div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\n=1(T21, 0) !=? =1(X9, X9)\n=1(T20, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: =1(T23, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: =1(T23, 0), SCOPE: 6, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\n=1(T21, 0) !=? =1(X9, X9)\n=1(T20, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(minus(T39, T40, X43), ','(div(X43, T40, X44), =1(T42, s(X44))))\n\nT39 is ground\nT40 is ground\nX43 is free; \nX44 is free; \n=1(T40, 0) !=? =1(X9, X9)\n=1(T39, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: minus(T39, T40, X43)\n\nT39 is ground\nT40 is ground\nX43 is free; \n=1(T40, 0) !=? =1(X9, X9)\n=1(T39, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(div(T45, T40, X44), =1(T42, s(X44)))\n\nT40 is ground\nT45 is ground\nX44 is free; \n=1(T40, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: minus(T39, T40, X43), SCOPE: 7, CLAUSE: 0 | minus(T39, T40, X43), SCOPE: 7, CLAUSE: 1 | minus(T39, T40, X43), SCOPE: 7, CLAUSE: 2\n\nT39 is ground\nT40 is ground\nX43 is free; \n=1(T40, 0) !=? =1(X9, X9)\n=1(T39, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(=1(T52, 0), ','(!_7, =1(X64, 0))) | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 1 | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX43 is free; \nX64 is free; \n=1(T53, 0) !=? =1(X9, X9)\n=1(T52, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(=1(T52, 0), ','(!_7, =1(X64, 0))), SCOPE: 8, CLAUSE: 6 | ?_8 | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 1 | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX43 is free; \nX64 is free; \n=1(T53, 0) !=? =1(X9, X9)\n=1(T52, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ?_8 | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 1 | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX43 is free; \n=1(T53, 0) !=? =1(X9, X9)\n=1(T52, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: minus(T52, T53, X43), SCOPE: 7, CLAUSE: 1 | minus(T52, T53, X43), SCOPE: 7, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX43 is free; \n=1(T53, 0) !=? =1(X9, X9)\n=1(T52, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(=1(T60, 0), ','(!_7, =1(X77, T59))) | minus(T59, T60, X43), SCOPE: 7, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX43 is free; \nX77 is free; \n=1(T60, 0) !=? =1(X9, X9)\n=1(T59, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(=1(T60, 0), ','(!_7, =1(X77, T59))), SCOPE: 9, CLAUSE: 6 | ?_9 | minus(T59, T60, X43), SCOPE: 7, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX43 is free; \nX77 is free; \n=1(T60, 0) !=? =1(X9, X9)\n=1(T59, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ?_9 | minus(T59, T60, X43), SCOPE: 7, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX43 is free; \n=1(T60, 0) !=? =1(X9, X9)\n=1(T59, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: minus(T59, T60, X43), SCOPE: 7, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX43 is free; \n=1(T60, 0) !=? =1(X9, X9)\n=1(T59, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(=1(T66, s(X98)), ','(=1(T67, s(X99)), minus(X98, X99, X100)))\n\nT66 is ground\nT67 is ground\nX100 is free; \nX98 is free; \nX99 is free; \n=1(T67, 0) !=? =1(X9, X9)\n=1(T66, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(=1(T66, s(X98)), ','(=1(T67, s(X99)), minus(X98, X99, X100))), SCOPE: 10, CLAUSE: 6\n\nT66 is ground\nT67 is ground\nX100 is free; \nX98 is free; \nX99 is free; \n=1(T67, 0) !=? =1(X9, X9)\n=1(T66, 0) !=? =1(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(=1(T67, s(X99)), minus(T73, X99, X100))\n\nT67 is ground\nT73 is ground\nX100 is free; \nX99 is free; \n=1(T67, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(=1(T67, s(X99)), minus(T73, X99, X100)), SCOPE: 11, CLAUSE: 6\n\nT67 is ground\nT73 is ground\nX100 is free; \nX99 is free; \n=1(T67, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: minus(T73, T80, X100)\n\nT73 is ground\nT80 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: minus(T73, T80, X100), SCOPE: 12, CLAUSE: 0 | minus(T73, T80, X100), SCOPE: 12, CLAUSE: 1 | minus(T73, T80, X100), SCOPE: 12, CLAUSE: 2\n\nT73 is ground\nT80 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(=1(T87, 0), ','(!_12, =1(X135, 0))) | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 1 | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX100 is free; \nX135 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(=1(T87, 0), ','(!_12, =1(X135, 0))), SCOPE: 13, CLAUSE: 6 | ?_13 | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 1 | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX100 is free; \nX135 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(!_12, =1(X135, 0)) | ?_13 | minus(0, T88, X100), SCOPE: 12, CLAUSE: 1 | minus(0, T88, X100), SCOPE: 12, CLAUSE: 2\n\nT88 is ground\nX100 is free; \nX135 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ?_13 | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 1 | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX100 is free; \n=1(T87, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: =1(X135, 0)\n\nX135 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: =1(X135, 0), SCOPE: 14, CLAUSE: 6\n\nX135 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: minus(T87, T88, X100), SCOPE: 12, CLAUSE: 1 | minus(T87, T88, X100), SCOPE: 12, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX100 is free; \n=1(T87, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(=1(T97, 0), ','(!_12, =1(X157, T96))) | minus(T96, T97, X100), SCOPE: 12, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX100 is free; \nX157 is free; \n=1(T96, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(=1(T97, 0), ','(!_12, =1(X157, T96))), SCOPE: 15, CLAUSE: 6 | ?_15 | minus(T96, T97, X100), SCOPE: 12, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX100 is free; \nX157 is free; \n=1(T96, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(!_12, =1(X157, T96)) | ?_15 | minus(T96, 0, X100), SCOPE: 12, CLAUSE: 2\n\nT96 is ground\nX100 is free; \nX157 is free; \n=1(T96, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ?_15 | minus(T96, T97, X100), SCOPE: 12, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX100 is free; \n=1(T96, 0) !=? =1(X138, X138)\n=1(T97, 0) !=? =1(X160, X160)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: =1(X157, T96)\n\nT96 is ground\nX157 is free; \n=1(T96, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: =1(X157, T96), SCOPE: 16, CLAUSE: 6\n\nT96 is ground\nX157 is free; \n=1(T96, 0) !=? =1(X138, X138)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: minus(T96, T97, X100), SCOPE: 12, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX100 is free; \n=1(T96, 0) !=? =1(X138, X138)\n=1(T97, 0) !=? =1(X160, X160)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(=1(T108, s(X187)), ','(=1(T109, s(X188)), minus(X187, X188, X189)))\n\nT108 is ground\nT109 is ground\nX189 is free; \nX187 is free; \nX188 is free; \n=1(T108, 0) !=? =1(X138, X138)\n=1(T109, 0) !=? =1(X160, X160)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(=1(T108, s(X187)), ','(=1(T109, s(X188)), minus(X187, X188, X189))), SCOPE: 17, CLAUSE: 6\n\nT108 is ground\nT109 is ground\nX189 is free; \nX187 is free; \nX188 is free; \n=1(T108, 0) !=? =1(X138, X138)\n=1(T109, 0) !=? =1(X160, X160)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(=1(T109, s(X188)), minus(T115, X188, X189))\n\nT109 is ground\nT115 is ground\nX189 is free; \nX188 is free; \n=1(T109, 0) !=? =1(X160, X160)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(=1(T109, s(X188)), minus(T115, X188, X189)), SCOPE: 18, CLAUSE: 6\n\nT109 is ground\nT115 is ground\nX189 is free; \nX188 is free; \n=1(T109, 0) !=? =1(X160, X160)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: minus(T115, T122, X189)\n\nT115 is ground\nT122 is ground\nX189 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: div(T45, T40, X44)\n\nT40 is ground\nT45 is ground\nX44 is free; \n=1(T40, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: =1(T42, s(T127))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: div(T45, T40, X44), SCOPE: 19, CLAUSE: 3 | div(T45, T40, X44), SCOPE: 19, CLAUSE: 4 | div(T45, T40, X44), SCOPE: 19, CLAUSE: 5\n\nT40 is ground\nT45 is ground\nX44 is free; \n=1(T40, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(=1(T135, 0), ','(!_19, user_defined_fail)) | div(T134, T135, X44), SCOPE: 19, CLAUSE: 4 | div(T134, T135, X44), SCOPE: 19, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX44 is free; \n=1(T135, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(=1(T135, 0), ','(!_19, user_defined_fail)), SCOPE: 20, CLAUSE: 6 | ?_20 | div(T134, T135, X44), SCOPE: 19, CLAUSE: 4 | div(T134, T135, X44), SCOPE: 19, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX44 is free; \n=1(T135, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ?_20 | div(T134, T135, X44), SCOPE: 19, CLAUSE: 4 | div(T134, T135, X44), SCOPE: 19, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX44 is free; \n=1(T135, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: div(T134, T135, X44), SCOPE: 19, CLAUSE: 4 | div(T134, T135, X44), SCOPE: 19, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX44 is free; \n=1(T135, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(=1(T141, 0), ','(!_19, =1(X245, 0))) | div(T141, T142, X44), SCOPE: 19, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX44 is free; \nX245 is free; \n=1(T142, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(=1(T141, 0), ','(!_19, =1(X245, 0))), SCOPE: 21, CLAUSE: 6 | ?_21 | div(T141, T142, X44), SCOPE: 19, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX44 is free; \nX245 is free; \n=1(T142, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(!_19, =1(X245, 0)) | ?_21 | div(0, T142, X44), SCOPE: 19, CLAUSE: 5\n\nT142 is ground\nX44 is free; \nX245 is free; \n=1(T142, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ?_21 | div(T141, T142, X44), SCOPE: 19, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX44 is free; \n=1(T142, 0) !=? =1(X9, X9)\n=1(T141, 0) !=? =1(X248, X248)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: =1(X245, 0)\n\nX245 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: =1(X245, 0), SCOPE: 22, CLAUSE: 6\n\nX245 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: div(T141, T142, X44), SCOPE: 19, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX44 is free; \n=1(T142, 0) !=? =1(X9, X9)\n=1(T141, 0) !=? =1(X248, X248)", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(minus(T152, T153, X273), ','(div(X273, T153, X274), =1(X275, s(X274))))\n\nT152 is ground\nT153 is ground\nX275 is free; \nX273 is free; \nX274 is free; \n=1(T153, 0) !=? =1(X9, X9)\n=1(T152, 0) !=? =1(X248, X248)", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: minus(T152, T153, X273)\n\nT152 is ground\nT153 is ground\nX273 is free; \n=1(T153, 0) !=? =1(X9, X9)\n=1(T152, 0) !=? =1(X248, X248)", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(div(T156, T153, X274), =1(X275, s(X274)))\n\nT153 is ground\nT156 is ground\nX275 is free; \nX274 is free; \n=1(T153, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: div(T156, T153, X274)\n\nT153 is ground\nT156 is ground\nX274 is free; \n=1(T153, 0) !=? =1(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: =1(X275, s(T161))\n\nX275 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: =1(X275, s(T161)), SCOPE: 23, CLAUSE: 6\n\nX275 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: =1(T42, s(T127)), SCOPE: 24, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 96 [arrowhead = none ];
96 -> 2;

97 [label="ONLY EVAL with clause\ndiv(X4, X5, X6) :- ','(=1(X5, 0), ','(!_1, user_defined_fail)).\nand substitution\nT1 -> T7\nX4 -> T7\nT2 -> T8\nX5 -> T8\nT3 -> T9\nX6 -> T9", fontsize=14, style = filled, fillcolor = white];
2 -> 97 [arrowhead = none ];
97 -> 3;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 98 [arrowhead = none ];
98 -> 4;

99 [label="EVAL with clause\n=1(X9, X9).\nand substitution\nT8 -> 0\nX9 -> 0\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 99 [arrowhead = none ];
99 -> 5;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 100 [arrowhead = none ];
100 -> 6;

101 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 101 [arrowhead = none ];
101 -> 7;

102 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 102 [arrowhead = none ];
102 -> 12;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 103 [arrowhead = none ];
103 -> 8;

104 [label="ONLY EVAL with clause\nuser_defined_fail :- fail(b).\nand substitution", fontsize=14, style = filled, fillcolor = white];
8 -> 104 [arrowhead = none ];
104 -> 9;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 105 [arrowhead = none ];
105 -> 10;

106 [label="BACKTRACK\nfor clause: fail(a)because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 106 [arrowhead = none ];
106 -> 11;

107 [label="ONLY EVAL with clause\ndiv(X17, X18, X19) :- ','(=1(X17, 0), ','(!_1, =1(X19, 0))).\nand substitution\nT7 -> T20\nX17 -> T20\nT8 -> T21\nX18 -> T21\nT3 -> T23\nX19 -> T23\nT22 -> T23", fontsize=14, style = filled, fillcolor = white];
12 -> 107 [arrowhead = none ];
107 -> 13;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 108 [arrowhead = none ];
108 -> 14;

109 [label="EVAL with clause\n=1(X22, X22).\nand substitution\nT20 -> 0\nX22 -> 0\nT26 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 109 [arrowhead = none ];
109 -> 15;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 110 [arrowhead = none ];
110 -> 16;

111 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 111 [arrowhead = none ];
111 -> 17;

112 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 112 [arrowhead = none ];
112 -> 22;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 113 [arrowhead = none ];
113 -> 18;

114 [label="EVAL with clause\n=1(X26, X26).\nand substitution\nT23 -> 0\nX26 -> 0\nT29 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 114 [arrowhead = none ];
114 -> 19;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 115 [arrowhead = none ];
115 -> 20;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 116 [arrowhead = none ];
116 -> 21;

117 [label="ONLY EVAL with clause\ndiv(X40, X41, X42) :- ','(minus(X40, X41, X43), ','(div(X43, X41, X44), =1(X42, s(X44)))).\nand substitution\nT20 -> T39\nX40 -> T39\nT21 -> T40\nX41 -> T40\nT3 -> T42\nX42 -> T42\nT41 -> T42", fontsize=14, style = filled, fillcolor = white];
22 -> 117 [arrowhead = none ];
117 -> 23;

118 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 118 [arrowhead = none ];
118 -> 24;

119 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nreplacements:\nX43 -> T45", fontsize=14, style = filled, fillcolor = violet];
23 -> 119 [arrowhead = none ];
119 -> 25;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 120 [arrowhead = none ];
120 -> 26;

121 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 121 [arrowhead = none ];
121 -> 68;

122 [label="SPLIT 2\nreplacements:\nX44 -> T127", fontsize=14, style = filled, fillcolor = violet];
25 -> 122 [arrowhead = none ];
122 -> 69;

123 [label="ONLY EVAL with clause\nminus(X61, X62, X63) :- ','(=1(X61, 0), ','(!_7, =1(X63, 0))).\nand substitution\nT39 -> T52\nX61 -> T52\nT40 -> T53\nX62 -> T53\nX43 -> X64\nX63 -> X64", fontsize=14, style = filled, fillcolor = white];
26 -> 123 [arrowhead = none ];
123 -> 27;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 124 [arrowhead = none ];
124 -> 28;

125 [label="BACKTRACK\nfor clause: =1(X, X)\nwith clash: (=1(T52, 0), =1(X22, X22))", fontsize=14, style = filled, fillcolor = white];
28 -> 125 [arrowhead = none ];
125 -> 29;

126 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 126 [arrowhead = none ];
126 -> 30;

127 [label="ONLY EVAL with clause\nminus(X74, X75, X76) :- ','(=1(X75, 0), ','(!_7, =1(X76, X74))).\nand substitution\nT52 -> T59\nX74 -> T59\nT53 -> T60\nX75 -> T60\nX43 -> X77\nX76 -> X77", fontsize=14, style = filled, fillcolor = white];
30 -> 127 [arrowhead = none ];
127 -> 31;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 128 [arrowhead = none ];
128 -> 32;

129 [label="BACKTRACK\nfor clause: =1(X, X)\nwith clash: (=1(T60, 0), =1(X9, X9))", fontsize=14, style = filled, fillcolor = white];
32 -> 129 [arrowhead = none ];
129 -> 33;

130 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 130 [arrowhead = none ];
130 -> 34;

131 [label="ONLY EVAL with clause\nminus(X95, X96, X97) :- ','(=1(X95, s(X98)), ','(=1(X96, s(X99)), minus(X98, X99, X97))).\nand substitution\nT59 -> T66\nX95 -> T66\nT60 -> T67\nX96 -> T67\nX43 -> X100\nX97 -> X100", fontsize=14, style = filled, fillcolor = white];
34 -> 131 [arrowhead = none ];
131 -> 35;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 132 [arrowhead = none ];
132 -> 36;

133 [label="EVAL with clause\n=1(X109, X109).\nand substitution\nT66 -> s(T73)\nX109 -> s(T73)\nX98 -> T73\nT72 -> s(T73)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 133 [arrowhead = none ];
133 -> 37;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 134 [arrowhead = none ];
134 -> 38;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 135 [arrowhead = none ];
135 -> 39;

136 [label="EVAL with clause\n=1(X118, X118).\nand substitution\nT67 -> s(T80)\nX118 -> s(T80)\nX99 -> T80\nT79 -> s(T80)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 136 [arrowhead = none ];
136 -> 40;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 137 [arrowhead = none ];
137 -> 41;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 138 [arrowhead = none ];
138 -> 42;

139 [label="ONLY EVAL with clause\nminus(X132, X133, X134) :- ','(=1(X132, 0), ','(!_12, =1(X134, 0))).\nand substitution\nT73 -> T87\nX132 -> T87\nT80 -> T88\nX133 -> T88\nX100 -> X135\nX134 -> X135", fontsize=14, style = filled, fillcolor = white];
42 -> 139 [arrowhead = none ];
139 -> 43;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 140 [arrowhead = none ];
140 -> 44;

141 [label="EVAL with clause\n=1(X138, X138).\nand substitution\nT87 -> 0\nX138 -> 0\nT91 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 141 [arrowhead = none ];
141 -> 45;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 142 [arrowhead = none ];
142 -> 46;

143 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 143 [arrowhead = none ];
143 -> 47;

144 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 144 [arrowhead = none ];
144 -> 51;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 145 [arrowhead = none ];
145 -> 48;

146 [label="ONLY EVAL with clause\n=1(X144, X144).\nand substitution\nX135 -> 0\nX144 -> 0\nX145 -> 0", fontsize=14, style = filled, fillcolor = white];
48 -> 146 [arrowhead = none ];
146 -> 49;

147 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 147 [arrowhead = none ];
147 -> 50;

148 [label="ONLY EVAL with clause\nminus(X154, X155, X156) :- ','(=1(X155, 0), ','(!_12, =1(X156, X154))).\nand substitution\nT87 -> T96\nX154 -> T96\nT88 -> T97\nX155 -> T97\nX100 -> X157\nX156 -> X157", fontsize=14, style = filled, fillcolor = white];
51 -> 148 [arrowhead = none ];
148 -> 52;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 149 [arrowhead = none ];
149 -> 53;

150 [label="EVAL with clause\n=1(X160, X160).\nand substitution\nT97 -> 0\nX160 -> 0\nT100 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 150 [arrowhead = none ];
150 -> 54;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 151 [arrowhead = none ];
151 -> 55;

152 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 152 [arrowhead = none ];
152 -> 56;

153 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 153 [arrowhead = none ];
153 -> 60;

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 154 [arrowhead = none ];
154 -> 57;

155 [label="ONLY EVAL with clause\n=1(X166, X166).\nand substitution\nX157 -> T103\nX166 -> T103\nT96 -> T103\nX167 -> T103", fontsize=14, style = filled, fillcolor = white];
57 -> 155 [arrowhead = none ];
155 -> 58;

156 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 156 [arrowhead = none ];
156 -> 59;

157 [label="ONLY EVAL with clause\nminus(X184, X185, X186) :- ','(=1(X184, s(X187)), ','(=1(X185, s(X188)), minus(X187, X188, X186))).\nand substitution\nT96 -> T108\nX184 -> T108\nT97 -> T109\nX185 -> T109\nX100 -> X189\nX186 -> X189", fontsize=14, style = filled, fillcolor = white];
60 -> 157 [arrowhead = none ];
157 -> 61;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 158 [arrowhead = none ];
158 -> 62;

159 [label="EVAL with clause\n=1(X198, X198).\nand substitution\nT108 -> s(T115)\nX198 -> s(T115)\nX187 -> T115\nT114 -> s(T115)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 159 [arrowhead = none ];
159 -> 63;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 160 [arrowhead = none ];
160 -> 64;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 161 [arrowhead = none ];
161 -> 65;

162 [label="EVAL with clause\n=1(X207, X207).\nand substitution\nT109 -> s(T122)\nX207 -> s(T122)\nX188 -> T122\nT121 -> s(T122)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 162 [arrowhead = none ];
162 -> 66;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 163 [arrowhead = none ];
163 -> 67;

164 [label="INSTANCE with matching:\nT73 -> T115\nT80 -> T122\nX100 -> X189", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 164 [arrowhead = none , style=dashed];
164 -> 40[style=dashed];

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 165 [arrowhead = none ];
165 -> 70;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 166 [arrowhead = none ];
166 -> 92;

167 [label="ONLY EVAL with clause\ndiv(X229, X230, X231) :- ','(=1(X230, 0), ','(!_19, user_defined_fail)).\nand substitution\nT45 -> T134\nX229 -> T134\nT40 -> T135\nX230 -> T135\nX44 -> X232\nX231 -> X232", fontsize=14, style = filled, fillcolor = white];
70 -> 167 [arrowhead = none ];
167 -> 71;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 168 [arrowhead = none ];
168 -> 72;

169 [label="BACKTRACK\nfor clause: =1(X, X)\nwith clash: (=1(T135, 0), =1(X9, X9))", fontsize=14, style = filled, fillcolor = white];
72 -> 169 [arrowhead = none ];
169 -> 73;

170 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 170 [arrowhead = none ];
170 -> 74;

171 [label="ONLY EVAL with clause\ndiv(X242, X243, X244) :- ','(=1(X242, 0), ','(!_19, =1(X244, 0))).\nand substitution\nT134 -> T141\nX242 -> T141\nT135 -> T142\nX243 -> T142\nX44 -> X245\nX244 -> X245", fontsize=14, style = filled, fillcolor = white];
74 -> 171 [arrowhead = none ];
171 -> 75;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 172 [arrowhead = none ];
172 -> 76;

173 [label="EVAL with clause\n=1(X248, X248).\nand substitution\nT141 -> 0\nX248 -> 0\nT145 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 173 [arrowhead = none ];
173 -> 77;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 174 [arrowhead = none ];
174 -> 78;

175 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 175 [arrowhead = none ];
175 -> 79;

176 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 176 [arrowhead = none ];
176 -> 83;

177 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 177 [arrowhead = none ];
177 -> 80;

178 [label="ONLY EVAL with clause\n=1(X254, X254).\nand substitution\nX245 -> 0\nX254 -> 0\nX255 -> 0", fontsize=14, style = filled, fillcolor = white];
80 -> 178 [arrowhead = none ];
178 -> 81;

179 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 179 [arrowhead = none ];
179 -> 82;

180 [label="ONLY EVAL with clause\ndiv(X270, X271, X272) :- ','(minus(X270, X271, X273), ','(div(X273, X271, X274), =1(X272, s(X274)))).\nand substitution\nT141 -> T152\nX270 -> T152\nT142 -> T153\nX271 -> T153\nX44 -> X275\nX272 -> X275", fontsize=14, style = filled, fillcolor = white];
83 -> 180 [arrowhead = none ];
180 -> 84;

181 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
84 -> 181 [arrowhead = none ];
181 -> 85;

182 [label="SPLIT 2\nnew knowledge:\nT156 is ground\nreplacements:\nX273 -> T156", fontsize=14, style = filled, fillcolor = violet];
84 -> 182 [arrowhead = none ];
182 -> 86;

183 [label="INSTANCE with matching:\nT39 -> T152\nT40 -> T153\nX43 -> X273\nX22 -> X248", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 183 [arrowhead = none , style=dashed];
183 -> 24[style=dashed];

184 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
86 -> 184 [arrowhead = none ];
184 -> 87;

185 [label="SPLIT 2\nreplacements:\nX274 -> T161", fontsize=14, style = filled, fillcolor = violet];
86 -> 185 [arrowhead = none ];
185 -> 88;

186 [label="INSTANCE with matching:\nT45 -> T156\nT40 -> T153\nX44 -> X274", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 186 [arrowhead = none , style=dashed];
186 -> 68[style=dashed];

187 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 187 [arrowhead = none ];
187 -> 89;

188 [label="ONLY EVAL with clause\n=1(X296, X296).\nand substitution\nX275 -> s(T166)\nX296 -> s(T166)\nT161 -> T166\nX297 -> s(T166)", fontsize=14, style = filled, fillcolor = white];
89 -> 188 [arrowhead = none ];
188 -> 90;

189 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 189 [arrowhead = none ];
189 -> 91;

190 [label="EVAL with clause\n=1(X300, X300).\nand substitution\nT42 -> s(T172)\nX300 -> s(T172)\nT127 -> T172\nT171 -> s(T172)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 190 [arrowhead = none ];
190 -> 93;

191 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 191 [arrowhead = none ];
191 -> 94;

192 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 192 [arrowhead = none ];
192 -> 95;

}

(6) Obligation:

Triples:

minus40(s(T115), s(T122), X189) :- minus40(T115, T122, X189).
minus24(s(T73), s(T80), X100) :- minus40(T73, T80, X100).
div68(T152, T153, X275) :- minus24(T152, T153, X273).
div68(T152, T153, X275) :- ','(minusc24(T152, T153, T156), div68(T156, T153, X274)).
div1(T39, T40, T42) :- minus24(T39, T40, X43).
div1(T39, T40, T42) :- ','(minusc24(T39, T40, T45), div68(T45, T40, X44)).

Clauses:

minusc40(0, T88, 0).
minusc40(T103, 0, T103).
minusc40(s(T115), s(T122), X189) :- minusc40(T115, T122, X189).
minusc24(s(T73), s(T80), X100) :- minusc40(T73, T80, X100).
divc68(0, T142, 0).
divc68(T152, T153, s(T166)) :- ','(minusc24(T152, T153, T156), divc68(T156, T153, T166)).

Afs:

div1(x1, x2, x3)  =  div1(x1, x2)

(7) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
div1_in: (b,b,f)
minus24_in: (b,b,f)
minus40_in: (b,b,f)
minusc24_in: (b,b,f)
minusc40_in: (b,b,f)
div68_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

DIV1_IN_GGA(T39, T40, T42) → U6_GGA(T39, T40, T42, minus24_in_gga(T39, T40, X43))
DIV1_IN_GGA(T39, T40, T42) → MINUS24_IN_GGA(T39, T40, X43)
MINUS24_IN_GGA(s(T73), s(T80), X100) → U2_GGA(T73, T80, X100, minus40_in_gga(T73, T80, X100))
MINUS24_IN_GGA(s(T73), s(T80), X100) → MINUS40_IN_GGA(T73, T80, X100)
MINUS40_IN_GGA(s(T115), s(T122), X189) → U1_GGA(T115, T122, X189, minus40_in_gga(T115, T122, X189))
MINUS40_IN_GGA(s(T115), s(T122), X189) → MINUS40_IN_GGA(T115, T122, X189)
DIV1_IN_GGA(T39, T40, T42) → U7_GGA(T39, T40, T42, minusc24_in_gga(T39, T40, T45))
U7_GGA(T39, T40, T42, minusc24_out_gga(T39, T40, T45)) → U8_GGA(T39, T40, T42, div68_in_gga(T45, T40, X44))
U7_GGA(T39, T40, T42, minusc24_out_gga(T39, T40, T45)) → DIV68_IN_GGA(T45, T40, X44)
DIV68_IN_GGA(T152, T153, X275) → U3_GGA(T152, T153, X275, minus24_in_gga(T152, T153, X273))
DIV68_IN_GGA(T152, T153, X275) → MINUS24_IN_GGA(T152, T153, X273)
DIV68_IN_GGA(T152, T153, X275) → U4_GGA(T152, T153, X275, minusc24_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X275, minusc24_out_gga(T152, T153, T156)) → U5_GGA(T152, T153, X275, div68_in_gga(T156, T153, X274))
U4_GGA(T152, T153, X275, minusc24_out_gga(T152, T153, T156)) → DIV68_IN_GGA(T156, T153, X274)

The TRS R consists of the following rules:

minusc24_in_gga(s(T73), s(T80), X100) → U11_gga(T73, T80, X100, minusc40_in_gga(T73, T80, X100))
minusc40_in_gga(0, T88, 0) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0, T103) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122), X189) → U10_gga(T115, T122, X189, minusc40_in_gga(T115, T122, X189))
U10_gga(T115, T122, X189, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)
U11_gga(T73, T80, X100, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)

The argument filtering Pi contains the following mapping:
minus24_in_gga(x1, x2, x3)  =  minus24_in_gga(x1, x2)
s(x1)  =  s(x1)
minus40_in_gga(x1, x2, x3)  =  minus40_in_gga(x1, x2)
minusc24_in_gga(x1, x2, x3)  =  minusc24_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
minusc40_in_gga(x1, x2, x3)  =  minusc40_in_gga(x1, x2)
0  =  0
minusc40_out_gga(x1, x2, x3)  =  minusc40_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
minusc24_out_gga(x1, x2, x3)  =  minusc24_out_gga(x1, x2, x3)
div68_in_gga(x1, x2, x3)  =  div68_in_gga(x1, x2)
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
MINUS24_IN_GGA(x1, x2, x3)  =  MINUS24_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
MINUS40_IN_GGA(x1, x2, x3)  =  MINUS40_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
DIV68_IN_GGA(x1, x2, x3)  =  DIV68_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(8) Obligation:

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

DIV1_IN_GGA(T39, T40, T42) → U6_GGA(T39, T40, T42, minus24_in_gga(T39, T40, X43))
DIV1_IN_GGA(T39, T40, T42) → MINUS24_IN_GGA(T39, T40, X43)
MINUS24_IN_GGA(s(T73), s(T80), X100) → U2_GGA(T73, T80, X100, minus40_in_gga(T73, T80, X100))
MINUS24_IN_GGA(s(T73), s(T80), X100) → MINUS40_IN_GGA(T73, T80, X100)
MINUS40_IN_GGA(s(T115), s(T122), X189) → U1_GGA(T115, T122, X189, minus40_in_gga(T115, T122, X189))
MINUS40_IN_GGA(s(T115), s(T122), X189) → MINUS40_IN_GGA(T115, T122, X189)
DIV1_IN_GGA(T39, T40, T42) → U7_GGA(T39, T40, T42, minusc24_in_gga(T39, T40, T45))
U7_GGA(T39, T40, T42, minusc24_out_gga(T39, T40, T45)) → U8_GGA(T39, T40, T42, div68_in_gga(T45, T40, X44))
U7_GGA(T39, T40, T42, minusc24_out_gga(T39, T40, T45)) → DIV68_IN_GGA(T45, T40, X44)
DIV68_IN_GGA(T152, T153, X275) → U3_GGA(T152, T153, X275, minus24_in_gga(T152, T153, X273))
DIV68_IN_GGA(T152, T153, X275) → MINUS24_IN_GGA(T152, T153, X273)
DIV68_IN_GGA(T152, T153, X275) → U4_GGA(T152, T153, X275, minusc24_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X275, minusc24_out_gga(T152, T153, T156)) → U5_GGA(T152, T153, X275, div68_in_gga(T156, T153, X274))
U4_GGA(T152, T153, X275, minusc24_out_gga(T152, T153, T156)) → DIV68_IN_GGA(T156, T153, X274)

The TRS R consists of the following rules:

minusc24_in_gga(s(T73), s(T80), X100) → U11_gga(T73, T80, X100, minusc40_in_gga(T73, T80, X100))
minusc40_in_gga(0, T88, 0) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0, T103) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122), X189) → U10_gga(T115, T122, X189, minusc40_in_gga(T115, T122, X189))
U10_gga(T115, T122, X189, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)
U11_gga(T73, T80, X100, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)

The argument filtering Pi contains the following mapping:
minus24_in_gga(x1, x2, x3)  =  minus24_in_gga(x1, x2)
s(x1)  =  s(x1)
minus40_in_gga(x1, x2, x3)  =  minus40_in_gga(x1, x2)
minusc24_in_gga(x1, x2, x3)  =  minusc24_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
minusc40_in_gga(x1, x2, x3)  =  minusc40_in_gga(x1, x2)
0  =  0
minusc40_out_gga(x1, x2, x3)  =  minusc40_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
minusc24_out_gga(x1, x2, x3)  =  minusc24_out_gga(x1, x2, x3)
div68_in_gga(x1, x2, x3)  =  div68_in_gga(x1, x2)
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
MINUS24_IN_GGA(x1, x2, x3)  =  MINUS24_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
MINUS40_IN_GGA(x1, x2, x3)  =  MINUS40_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
DIV68_IN_GGA(x1, x2, x3)  =  DIV68_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 11 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MINUS40_IN_GGA(s(T115), s(T122), X189) → MINUS40_IN_GGA(T115, T122, X189)

The TRS R consists of the following rules:

minusc24_in_gga(s(T73), s(T80), X100) → U11_gga(T73, T80, X100, minusc40_in_gga(T73, T80, X100))
minusc40_in_gga(0, T88, 0) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0, T103) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122), X189) → U10_gga(T115, T122, X189, minusc40_in_gga(T115, T122, X189))
U10_gga(T115, T122, X189, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)
U11_gga(T73, T80, X100, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
minusc24_in_gga(x1, x2, x3)  =  minusc24_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
minusc40_in_gga(x1, x2, x3)  =  minusc40_in_gga(x1, x2)
0  =  0
minusc40_out_gga(x1, x2, x3)  =  minusc40_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
minusc24_out_gga(x1, x2, x3)  =  minusc24_out_gga(x1, x2, x3)
MINUS40_IN_GGA(x1, x2, x3)  =  MINUS40_IN_GGA(x1, x2)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MINUS40_IN_GGA(s(T115), s(T122), X189) → MINUS40_IN_GGA(T115, T122, X189)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MINUS40_IN_GGA(x1, x2, x3)  =  MINUS40_IN_GGA(x1, x2)

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

MINUS40_IN_GGA(s(T115), s(T122)) → MINUS40_IN_GGA(T115, T122)

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

(16) 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:

• MINUS40_IN_GGA(s(T115), s(T122)) → MINUS40_IN_GGA(T115, T122)
  The graph contains the following edges 1 > 1, 2 > 2

(17) YES

(18) Obligation:

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

DIV68_IN_GGA(T152, T153, X275) → U4_GGA(T152, T153, X275, minusc24_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X275, minusc24_out_gga(T152, T153, T156)) → DIV68_IN_GGA(T156, T153, X274)

The TRS R consists of the following rules:

minusc24_in_gga(s(T73), s(T80), X100) → U11_gga(T73, T80, X100, minusc40_in_gga(T73, T80, X100))
minusc40_in_gga(0, T88, 0) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0, T103) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122), X189) → U10_gga(T115, T122, X189, minusc40_in_gga(T115, T122, X189))
U10_gga(T115, T122, X189, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)
U11_gga(T73, T80, X100, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
minusc24_in_gga(x1, x2, x3)  =  minusc24_in_gga(x1, x2)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
minusc40_in_gga(x1, x2, x3)  =  minusc40_in_gga(x1, x2)
0  =  0
minusc40_out_gga(x1, x2, x3)  =  minusc40_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
minusc24_out_gga(x1, x2, x3)  =  minusc24_out_gga(x1, x2, x3)
DIV68_IN_GGA(x1, x2, x3)  =  DIV68_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, 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:

DIV68_IN_GGA(T152, T153) → U4_GGA(T152, T153, minusc24_in_gga(T152, T153))
U4_GGA(T152, T153, minusc24_out_gga(T152, T153, T156)) → DIV68_IN_GGA(T156, T153)

The TRS R consists of the following rules:

minusc24_in_gga(s(T73), s(T80)) → U11_gga(T73, T80, minusc40_in_gga(T73, T80))
minusc40_in_gga(0, T88) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122)) → U10_gga(T115, T122, minusc40_in_gga(T115, T122))
U10_gga(T115, T122, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)
U11_gga(T73, T80, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)

The set Q consists of the following terms:

minusc24_in_gga(x0, x1)
minusc40_in_gga(x0, x1)
U10_gga(x0, x1, x2)
U11_gga(x0, x1, x2)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U4_GGA(T152, T153, minusc24_out_gga(T152, T153, T156)) → DIV68_IN_GGA(T156, T153)

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

POL(0) = 0   
POL(DIV68_IN_GGA(x₁, x₂)) = 1 + x₁   
POL(U10_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(U11_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(U4_GGA(x₁, x₂, x₃)) = 1 + x₃   
POL(minusc24_in_gga(x₁, x₂)) = x₁   
POL(minusc24_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(minusc40_in_gga(x₁, x₂)) = x₁   
POL(minusc40_out_gga(x₁, x₂, x₃)) = x₃   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

minusc24_in_gga(s(T73), s(T80)) → U11_gga(T73, T80, minusc40_in_gga(T73, T80))
minusc40_in_gga(0, T88) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122)) → U10_gga(T115, T122, minusc40_in_gga(T115, T122))
U11_gga(T73, T80, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)
U10_gga(T115, T122, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)

(22) Obligation:

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

DIV68_IN_GGA(T152, T153) → U4_GGA(T152, T153, minusc24_in_gga(T152, T153))

The TRS R consists of the following rules:

minusc24_in_gga(s(T73), s(T80)) → U11_gga(T73, T80, minusc40_in_gga(T73, T80))
minusc40_in_gga(0, T88) → minusc40_out_gga(0, T88, 0)
minusc40_in_gga(T103, 0) → minusc40_out_gga(T103, 0, T103)
minusc40_in_gga(s(T115), s(T122)) → U10_gga(T115, T122, minusc40_in_gga(T115, T122))
U10_gga(T115, T122, minusc40_out_gga(T115, T122, X189)) → minusc40_out_gga(s(T115), s(T122), X189)
U11_gga(T73, T80, minusc40_out_gga(T73, T80, X100)) → minusc24_out_gga(s(T73), s(T80), X100)

The set Q consists of the following terms:

minusc24_in_gga(x0, x1)
minusc40_in_gga(x0, x1)
U10_gga(x0, x1, x2)
U11_gga(x0, x1, x2)

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