Left Termination of the query pattern fold_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

fold(X, [], X).
fold(X, Y, Z) :- ','(no(empty(Y)), ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z))))).
myop(a, b, c).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Queries:

fold(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | fold(T5, [], T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nfold(T1, T2, T3) !=? fold(X6, [], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: fold(T5, [], T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10)))))\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 2, CLAUSE: 8 | ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 2, CLAUSE: 9\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(call(empty([])), ','(!_2, ','(failure(a), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))))) | ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 2, CLAUSE: 9\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(empty([]), ','(!_2, ','(failure(a), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))))) | ?_3 | ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 2, CLAUSE: 9\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(empty([]), ','(!_2, ','(failure(a), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))))), SCOPE: 4, CLAUSE: 7 | ?_4 | ?_3 | ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 2, CLAUSE: 9\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(!_2, ','(failure(a), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10)))))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 2, CLAUSE: 9\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(failure(a), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10)))))\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(failure(a), ','(head([], X14), ','(tail([], X15), ','(myop(T8, X14, X16), fold(X16, X15, T10))))), SCOPE: 5, CLAUSE: 10\n\nT8 is ground\nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(no(empty(T16)), ','(head(T16, X28), ','(tail(T16, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18)))))\n\nT15 is ground\nT16 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T16, T3) !=? fold(X6, [], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty(T16)), ','(head(T16, X28), ','(tail(T16, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 8 | ','(no(empty(T16)), ','(head(T16, X28), ','(tail(T16, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT16 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T16, T3) !=? fold(X6, [], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(call(empty(T21)), ','(!_6, ','(failure(a), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))))) | ','(no(empty(T21)), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT21 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T21, T3) !=? fold(X6, [], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(empty(T21), ','(!_6, ','(failure(a), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))))) | ?_7 | ','(no(empty(T21)), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT21 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T21, T3) !=? fold(X6, [], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(empty(T21), ','(!_6, ','(failure(a), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))))), SCOPE: 8, CLAUSE: 7 | ?_8 | ?_7 | ','(no(empty(T21)), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT21 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T21, T3) !=? fold(X6, [], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_6, ','(failure(a), ','(head([], X28), ','(tail([], X29), ','(myop(T15, X28, X30), fold(X30, X29, T18)))))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], X28), ','(tail([], X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nX28 is free; \nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_8 | ?_7 | ','(no(empty(T21)), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT21 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T21, T3) !=? fold(X6, [], X6)\nempty(T21) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), ','(head([], X28), ','(tail([], X29), ','(myop(T15, X28, X30), fold(X30, X29, T18)))))\n\nT15 is ground\nX28 is free; \nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(failure(a), ','(head([], X28), ','(tail([], X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 9, CLAUSE: 10\n\nT15 is ground\nX28 is free; \nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ?_7 | ','(no(empty(T21)), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT21 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T21, T3) !=? fold(X6, [], X6)\nempty(T21) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(no(empty(T21)), ','(head(T21, X28), ','(tail(T21, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))), SCOPE: 6, CLAUSE: 9\n\nT15 is ground\nT21 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T21, T3) !=? fold(X6, [], X6)\nempty(T21) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(head(T26, X28), ','(tail(T26, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18))))\n\nT15 is ground\nT26 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T26, T3) !=? fold(X6, [], X6)\nempty(T26) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(head(T26, X28), ','(tail(T26, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18)))), SCOPE: 10, CLAUSE: 3 | ','(head(T26, X28), ','(tail(T26, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18)))), SCOPE: 10, CLAUSE: 4\n\nT15 is ground\nT26 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T26, T3) !=? fold(X6, [], X6)\nempty(T26) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T26, X28), ','(tail(T26, X29), ','(myop(T15, X28, X30), fold(X30, X29, T18)))), SCOPE: 10, CLAUSE: 4\n\nT15 is ground\nT26 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nfold(T15, T26, T3) !=? fold(X6, [], X6)\nempty(T26) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(tail(.(T32, T33), X29), ','(myop(T15, T32, X30), fold(X30, X29, T18)))\n\nT15 is ground\nT32 is ground\nT33 is ground\nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T32, T33), X29), ','(myop(T15, T32, X30), fold(X30, X29, T18))), SCOPE: 11, CLAUSE: 5 | ','(tail(.(T32, T33), X29), ','(myop(T15, T32, X30), fold(X30, X29, T18))), SCOPE: 11, CLAUSE: 6\n\nT15 is ground\nT32 is ground\nT33 is ground\nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail(.(T32, T33), X29), ','(myop(T15, T32, X30), fold(X30, X29, T18))), SCOPE: 11, CLAUSE: 6\n\nT15 is ground\nT32 is ground\nT33 is ground\nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(myop(T15, T41, X30), fold(X30, T42, T18))\n\nT15 is ground\nT41 is ground\nT42 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(myop(T15, T41, X30), fold(X30, T42, T18)), SCOPE: 12, CLAUSE: 2\n\nT15 is ground\nT41 is ground\nT42 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: fold(c, T42, T18)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: fold(c, T42, T18), SCOPE: 13, CLAUSE: 0 | fold(c, T42, T18), SCOPE: 13, CLAUSE: 1\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: fold(c, T42, T18), SCOPE: 13, CLAUSE: 0\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: fold(c, T42, T18), SCOPE: 13, CLAUSE: 1\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(no(empty(T51)), ','(head(T51, X91), ','(tail(T51, X92), ','(myop(c, X91, X93), fold(X93, X92, T53)))))\n\nT51 is ground\nX91 is free; \nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(no(empty(T51)), ','(head(T51, X91), ','(tail(T51, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 8 | ','(no(empty(T51)), ','(head(T51, X91), ','(tail(T51, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT51 is ground\nX91 is free; \nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(call(empty(T57)), ','(!_14, ','(failure(a), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))))) | ','(no(empty(T57)), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT57 is ground\nX91 is free; \nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(empty(T57), ','(!_14, ','(failure(a), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))))) | ?_15 | ','(no(empty(T57)), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT57 is ground\nX91 is free; \nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(empty(T57), ','(!_14, ','(failure(a), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))))), SCOPE: 16, CLAUSE: 7 | ?_16 | ?_15 | ','(no(empty(T57)), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT57 is ground\nX91 is free; \nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(!_14, ','(failure(a), ','(head([], X91), ','(tail([], X92), ','(myop(c, X91, X93), fold(X93, X92, T53)))))) | ?_16 | ?_15 | ','(no(empty([])), ','(head([], X91), ','(tail([], X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nX91 is free\nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ?_16 | ?_15 | ','(no(empty(T57)), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT57 is ground\nX91 is free; \nX92 is free; \nX93 is free; \nempty(T57) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(failure(a), ','(head([], X91), ','(tail([], X92), ','(myop(c, X91, X93), fold(X93, X92, T53)))))\n\nX91 is free\nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(head([], X91), ','(tail([], X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 17, CLAUSE: 10\n\nX91 is free\nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ?_15 | ','(no(empty(T57)), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT57 is ground\nX91 is free; \nX92 is free; \nX93 is free; \nempty(T57) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(no(empty(T57)), ','(head(T57, X91), ','(tail(T57, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))), SCOPE: 14, CLAUSE: 9\n\nT57 is ground\nX91 is free; \nX92 is free; \nX93 is free; \nempty(T57) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(head(T62, X91), ','(tail(T62, X92), ','(myop(c, X91, X93), fold(X93, X92, T53))))\n\nT62 is ground\nX91 is free; \nX92 is free; \nX93 is free; \nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(head(T62, X91), ','(tail(T62, X92), ','(myop(c, X91, X93), fold(X93, X92, T53)))), SCOPE: 18, CLAUSE: 3 | ','(head(T62, X91), ','(tail(T62, X92), ','(myop(c, X91, X93), fold(X93, X92, T53)))), SCOPE: 18, CLAUSE: 4\n\nT62 is ground\nX91 is free; \nX92 is free; \nX93 is free; \nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(head(T62, X91), ','(tail(T62, X92), ','(myop(c, X91, X93), fold(X93, X92, T53)))), SCOPE: 18, CLAUSE: 4\n\nT62 is ground\nX91 is free; \nX92 is free; \nX93 is free; \nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail(.(T68, T69), X92), ','(myop(c, T68, X93), fold(X93, X92, T53)))\n\nT68 is ground\nT69 is ground\nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail(.(T68, T69), X92), ','(myop(c, T68, X93), fold(X93, X92, T53))), SCOPE: 19, CLAUSE: 5 | ','(tail(.(T68, T69), X92), ','(myop(c, T68, X93), fold(X93, X92, T53))), SCOPE: 19, CLAUSE: 6\n\nT68 is ground\nT69 is ground\nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(.(T68, T69), X92), ','(myop(c, T68, X93), fold(X93, X92, T53))), SCOPE: 19, CLAUSE: 6\n\nT68 is ground\nT69 is ground\nX92 is free; \nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(myop(c, T77, X93), fold(X93, T78, T53))\n\nT77 is ground\nT78 is ground\nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(myop(c, T77, X93), fold(X93, T78, T53)), SCOPE: 20, CLAUSE: 2\n\nT77 is ground\nT78 is ground\nX93 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 66 [arrowhead = none ];
66 -> 2;

67 [label="EVAL with clause\nfold(X6, [], X6).\nand substitution\nT1 -> T5\nX6 -> T5\nT2 -> []\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 67 [arrowhead = none ];
67 -> 3;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 68 [arrowhead = none ];
68 -> 4;

69 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 69 [arrowhead = none ];
69 -> 5;

70 [label="ONLY EVAL with clause\nfold(X25, X26, X27) :- ','(no(empty(X26)), ','(head(X26, X28), ','(tail(X26, X29), ','(myop(X25, X28, X30), fold(X30, X29, X27))))).\nand substitution\nT1 -> T15\nX25 -> T15\nT2 -> T16\nX26 -> T16\nT3 -> T18\nX27 -> T18\nT17 -> T18", fontsize=14, style = filled, fillcolor = white];
4 -> 70 [arrowhead = none ];
70 -> 15;

71 [label="ONLY EVAL with clause\nfold(X11, X12, X13) :- ','(no(empty(X12)), ','(head(X12, X14), ','(tail(X12, X15), ','(myop(X11, X14, X16), fold(X16, X15, X13))))).\nand substitution\nT5 -> T8\nX11 -> T8\nX12 -> []\nT3 -> T10\nX13 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = white];
5 -> 71 [arrowhead = none ];
71 -> 6;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 72 [arrowhead = none ];
72 -> 7;

73 [label="ONLY EVAL with clause\nno(X19) :- ','(call(X19), ','(!_2, failure(a))).\nand substitution\nX19 -> empty([])", fontsize=14, style = filled, fillcolor = white];
7 -> 73 [arrowhead = none ];
73 -> 8;

74 [label="CALL", fontsize=14, style = filled, fillcolor = white];
8 -> 74 [arrowhead = none ];
74 -> 9;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = white];
10 -> 76 [arrowhead = none ];
76 -> 11;

77 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 77 [arrowhead = none ];
77 -> 12;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 78 [arrowhead = none ];
78 -> 13;

79 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 79 [arrowhead = none ];
79 -> 14;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 80 [arrowhead = none ];
80 -> 16;

81 [label="ONLY EVAL with clause\nno(X33) :- ','(call(X33), ','(!_6, failure(a))).\nand substitution\nT16 -> T21\nX33 -> empty(T21)", fontsize=14, style = filled, fillcolor = white];
16 -> 81 [arrowhead = none ];
81 -> 17;

82 [label="CALL", fontsize=14, style = filled, fillcolor = white];
17 -> 82 [arrowhead = none ];
82 -> 18;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 83 [arrowhead = none ];
83 -> 19;

84 [label="EVAL with clause\nempty([]).\nand substitution\nT21 -> []", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 84 [arrowhead = none ];
84 -> 20;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 85 [arrowhead = none ];
85 -> 21;

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 88 [arrowhead = none ];
88 -> 23;

89 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 89 [arrowhead = none ];
89 -> 24;

90 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 90 [arrowhead = none ];
90 -> 26;

91 [label="ONLY EVAL with clause\nno(X40).\nand substitution\nT21 -> T26\nX40 -> empty(T26)", fontsize=14, style = filled, fillcolor = white];
26 -> 91 [arrowhead = none ];
91 -> 27;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 92 [arrowhead = none ];
92 -> 28;

93 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T26), empty([]))", fontsize=14, style = filled, fillcolor = white];
28 -> 93 [arrowhead = none ];
93 -> 29;

94 [label="EVAL with clause\nhead(.(X49, X50), X49).\nand substitution\nX49 -> T32\nX50 -> T33\nT26 -> .(T32, T33)\nX28 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 95 [arrowhead = none ];
95 -> 31;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 96 [arrowhead = none ];
96 -> 32;

97 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
32 -> 97 [arrowhead = none ];
97 -> 33;

98 [label="ONLY EVAL with clause\ntail(.(X63, X64), X64).\nand substitution\nT32 -> T41\nX63 -> T41\nT33 -> T42\nX64 -> T42\nX29 -> T42", fontsize=14, style = filled, fillcolor = white];
33 -> 98 [arrowhead = none ];
98 -> 34;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 99 [arrowhead = none ];
99 -> 35;

100 [label="EVAL with clause\nmyop(a, b, c).\nand substitution\nT15 -> a\nT41 -> b\nX30 -> c", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 36;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 101 [arrowhead = none ];
101 -> 37;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 102 [arrowhead = none ];
102 -> 38;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 103 [arrowhead = none ];
103 -> 39;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 104 [arrowhead = none ];
104 -> 40;

105 [label="EVAL with clause\nfold(X75, [], X75).\nand substitution\nX75 -> c\nT42 -> []\nT18 -> c", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 105 [arrowhead = none ];
105 -> 41;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 106 [arrowhead = none ];
106 -> 42;

107 [label="ONLY EVAL with clause\nfold(X88, X89, X90) :- ','(no(empty(X89)), ','(head(X89, X91), ','(tail(X89, X92), ','(myop(X88, X91, X93), fold(X93, X92, X90))))).\nand substitution\nX88 -> c\nT42 -> T51\nX89 -> T51\nT18 -> T53\nX90 -> T53\nT52 -> T53", fontsize=14, style = filled, fillcolor = white];
40 -> 107 [arrowhead = none ];
107 -> 44;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 108 [arrowhead = none ];
108 -> 43;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 109 [arrowhead = none ];
109 -> 45;

110 [label="ONLY EVAL with clause\nno(X98) :- ','(call(X98), ','(!_14, failure(a))).\nand substitution\nT51 -> T57\nX98 -> empty(T57)", fontsize=14, style = filled, fillcolor = white];
45 -> 110 [arrowhead = none ];
110 -> 46;

111 [label="CALL", fontsize=14, style = filled, fillcolor = white];
46 -> 111 [arrowhead = none ];
111 -> 47;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 112 [arrowhead = none ];
112 -> 48;

113 [label="EVAL with clause\nempty([]).\nand substitution\nT57 -> []", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 113 [arrowhead = none ];
113 -> 49;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 114 [arrowhead = none ];
114 -> 50;

115 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 115 [arrowhead = none ];
115 -> 51;

116 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 116 [arrowhead = none ];
116 -> 54;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 118 [arrowhead = none ];
118 -> 53;

119 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 119 [arrowhead = none ];
119 -> 55;

120 [label="ONLY EVAL with clause\nno(X105).\nand substitution\nT57 -> T62\nX105 -> empty(T62)", fontsize=14, style = filled, fillcolor = white];
55 -> 120 [arrowhead = none ];
120 -> 56;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 121 [arrowhead = none ];
121 -> 57;

122 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T62), empty([]))", fontsize=14, style = filled, fillcolor = white];
57 -> 122 [arrowhead = none ];
122 -> 58;

123 [label="EVAL with clause\nhead(.(X114, X115), X114).\nand substitution\nX114 -> T68\nX115 -> T69\nT62 -> .(T68, T69)\nX91 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 123 [arrowhead = none ];
123 -> 59;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 124 [arrowhead = none ];
124 -> 60;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 125 [arrowhead = none ];
125 -> 61;

126 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
61 -> 126 [arrowhead = none ];
126 -> 62;

127 [label="ONLY EVAL with clause\ntail(.(X128, X129), X129).\nand substitution\nT68 -> T77\nX128 -> T77\nT69 -> T78\nX129 -> T78\nX92 -> T78", fontsize=14, style = filled, fillcolor = white];
62 -> 127 [arrowhead = none ];
127 -> 63;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 128 [arrowhead = none ];
128 -> 64;

129 [label="BACKTRACK\nfor clause: myop(a, b, c)because of non-unification", fontsize=14, style = filled, fillcolor = white];
64 -> 129 [arrowhead = none ];
129 -> 65;

}

(2) Obligation:

Triples:


Clauses:


Afs:

(3) TPisEmptyProof (EQUIVALENT transformation)

There are no more dependency triples. Hence, the dependency triple problem trivially terminates.

(4) YES